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+{"text":"\\section{Introduction}\nIn view of the possible creation of a quark-gluon-plasma \n(QGP) in high-energy heavy ion reactions at the SPS, RHIC and \nLHC accelerators, descriptions of the space-time evolution \nof the deconfined phase and its transition into the hadronic \nphase are very much called for. Ideally, the transport \nequations should be directly derived from the QCD-Lagrangian, \na goal which is not yet achieved. Instead a number \nof approximate formulations have been proposed. They are \neither derived from effective Lagrangians (like those by \nFriedberg Lee \\cite{book_fl} or Nambu Jona-Lasinio \\cite{NJL} (NJL)) \nor they are of Monte-Carlo cascade type \\cite{cascade} \nwith experimental input like \nstring-fragmentation functions and\/or cross sections. Both \napproaches have their merits. While the ones based on \neffective Lagrangians permit to study certain aspects of QCD \n(e.g. chiral symmetry) and their manifestation in the space-time\n evolution of a highly excited strongly interacting \nsystem, the phenomenological approaches are very helpful in \nthe interpretation of the data. \n\n   The approach presented in this paper is closer in spirit to \nthe transport theories based on effective Lagrangians.\nWe propose a Vlasov equation for the evolution of partons, where \nthe medium dependent mass is directly obtained from QCD, more precisely from \nthe results of lattice calculations.\nWe concentrate on the transport properties of \nthe QGP close to the phase transition, but do not describe \nhadron dynamics. The lattice calculations for QCD at finite \ntemperature show a phase transition at a critical temperature \n$T_c$ which separates the regime of the QGP from the one of \nthe hadrons. Two aspects are related to the phase transition: \nrestoration of chiral symmetry and confinement. They \nmanifest themselves in rapid temperature variations around \n$T_c$ of the chiral condensate and the Poliakov loop, respectively.  \nBoth aspect are contained in the results (often called \"data\") \nof lattice calculations. \n\nThe basic assumption of our approach is the quasi-particle \npicture, with an effective mass $m$, which in thermal\nequilibrium depends on the \ntemperature  $T$, and on\nspace time ($x$, $t$), in the non-equilibrium situation. We treat \nthe equilibrium case in Sec. II of our paper and show, how \n$m(T)$ can be obtained from results of the lattice calculations. \nSec. III  deals with the non-equilibrium case, where \n$m(x,t)$ is calculated from the same gap-equation as $m(T)$. We \nreport results of a numerical calculation for the expansion of \na parton plasma, which shows confinement. The relation to other models \nis given in Sec. IV. \n\n\n\n\\section{Thermodynamics of the deconfinement phase transition \nin a quasi-particle approach}\n\n\\label{sec_2}\n\n\n\nThe quasi-particle model is one of the most simple approximations to an\ninteracting many body system. A  system which\nconsists of particles with definite mass and interactions among them\nis replaced by\na system of non-interacting quasi-particles whose masses $m(T)$ are\nchosen so that the thermodynamics of the original system is best\napproximated. For the case of QCD we make an ansatz for the pressure\ndensity (thermodynamic potential) for the partons \\cite{goren}:\n\\begin{eqnarray}\n\\label{press_equ}\nP_{qp}(m_1,m_2,\\dots,T)& = &\n\\sum_i  g_i \\int \\frac{d^3p}{(2 \\pi)^3} \\frac{p^2}{3 E_i(p)}\nf_i(E_i(p)) \\nonumber \\\\\n& &  - V(m_1,m_2,\\dots) \\  ,\n\\end{eqnarray}\nwhere the sum runs over  the parton species $i$ with degeneracy factor \n$g_i$ and where $f_i(E_i(p))$ are Bose or Fermi distribution functions,\nwhich depend on the quasi-particle energies\n$E_i(p)=\\sqrt{p^2+m_i^2}$.\n $V(m_1,m_2,\\dots)$ describes the mean-field contribution to the\ndispersion relation \\cite{goren}. The potential density $V$\ncontributes nontrivially  to\nthe thermodynamic relations   whenever the\nmasses are  temperature dependent.\n\nThe masses $m_i$ appearing in the pressure are phenomenological parameters\nwhich have to be chosen so that the thermodynamic potential\n (\\ref{press_equ}) has a minimum,\n\\begin{equation}\n\\bigg(\\frac{\\partial P}{\\partial m_i} \\bigg)_T=0 \\ , \\ \\ i=1,2,\\dots \\ \\ ,\n\\end{equation}\nleading to :\n\\begin{equation}\n\\label{gap_equ}\n\\frac{\\partial V}{\\partial m_i} + \n g_i \\int \\frac{d^3p}{(2 \\pi)^3} \\frac{m_i}{E_i}\nf_i(E_i) = 0 \\ .\n\\end{equation}\nThese equations have the form of a gap equation for the masses $m_i(T)$\nprovided the potential density $V(m_1,m_2,\\dots)$ is given.\n Eq. (\\ref{gap_equ}) \n is equivalent to the consistency relations \n\\cite{goren,levai}  allowing to obtain \nthe energy density from the pressure (\\ref{press_equ}), \nin the form as for the ideal gas~:\n\\begin{eqnarray}\n\\label{ener_equ}\n\\epsilon_{qp}(T)& =& \\sum_i  g_i \\int \\frac{d^3p}{(2 \\pi)^3} E_i\nf_i(E_i) + V(T) \\nonumber \\\\\n&= & \\epsilon_{kin}(m,T)+V(T) \\  ,\n\\end{eqnarray}\nwhere $V(T)=V(m_1(T),m_2(T),\\dots)$.\n\n\\begin{figure}[] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{f1.eps}\n\\caption[]{The energy density of the four-flavor QCD\ndivided by $T^4$ as a function of the scaled temperature $T\/T_c$.\n The points are from the \nlattice data \\cite{lattice}. The solid line is the result of the \nquasiparticle model described in the text with the chiral symmetry\nrestoration (case (b) and solid line in Fig. 2).\n The short-dashed and the dotted \nlines are the corresponding  kinetic\nenergy density and potential energy density respectively.\nThe dashed line represents the energy density obtained as a\nparameterization\nof the data points and leading to the parton mass increasing at high\ntemperature, (case (a) and dashed line in Fig. 2).\n The dashed-dotted line indicates the Stefan-Boltzmann limit.}\n\\label{lat_fig}\n\\end{center} \n\\end{figure}\n\nWe want to model the deconfinement phase transition of QCD in a\nquasi-particle approach and therefore have  to set a criterion how to\ndetermine the quasi-particle masses $m_i(T)$ or equivalently the mean-field\nenergy density $V(T)$. To date, the most detailed information about the\nQCD phase transition comes from the lattice calculations which give\nvarious thermodynamic functions, among them the energy density\n$\\epsilon_{lat}(T)$. We define  our quasi-particle\napproach by the requirement\n\\begin{equation}\n\\label{fit_equ}\n \\epsilon_{qp}(T)=\\epsilon_{lat}(T) \\ .\n\\end{equation}\nSince Eq. (\\ref{fit_equ}) is only one relation, only one function $m(T)$\ncan be determined. We therefore have to assume that our quasi-particle\nsystem consists of just one kind of massive partons ($i=1$ in Eqs. \n(\\ref{press_equ}) to (\\ref{ener_equ})). We will assume a Boltzmann type\ndistribution function $f(E,T)=\\exp^{-\\beta E}$.\nThe right hand side in Eq. (\\ref{fit_equ})\nbeing given, this equation is a constraint on the unknown\nfunction $m(T)$ in $\\epsilon_{qp}(T)$.\nWe note that  the quasi-particle model constrained by Eq. (\\ref{fit_equ})\n describes exactly the energy density of lattice QCD,\nbut may reproduce other thermodynamic function only approximately \n\\cite{goren,levai} which is not a surprise since lattice QCD is not\na quasi-particle gas. \nThe functional form for $m(T)$ can be obtained by\nsolving the differential equation\n\\begin{eqnarray}\n\\label{diff_equ}\n\\frac{dm(T)}{dT}&= &\\bigg({\\frac{d\\epsilon_{lat}(T)}{dT}-\\frac{\\partial \n\\epsilon_{kin}(m,T)}\n{\\partial T}}\\bigg) \\nonumber \\\\\n& & \/\\bigg({\\frac{\\partial \\epsilon_{kin}(m,T)}\n{\\partial m}+\\frac{dV}{dm}}\\bigg) \\ ,\n\\end{eqnarray}\nwhich is obtained from Eq. (\\ref{fit_equ}) by differentiating both sides\nwith respect to $T$ and\nwhere ${dV}\/{dm}$ is given by the gap equation (\\ref{gap_equ}).\nEq. (\\ref{diff_equ}) is a first order differential equation which determines \n$m(T)$ for a given energy density $\\epsilon_{lat}(T)$ and for an initial\nvalue $m(T_0)$. Then $V(T)$ can be obtained from\n$V(T) = \\epsilon_{lat}(T)- \\epsilon_{kin}(m(T),T)$.\n The lattice data are normalized so that\n$\\epsilon_{lat}(T)\n\\rightarrow 0$ for $T \\rightarrow 0$. It requires that $V(T)\\rightarrow 0$\nat low temperatures. \nIf $m(T)$ is not obtained from Eq. \\ref{diff_equ} but is\n given from some other considerations,\nthe gap equation can be integrated to obtain the potential $V(T)$~:\n\\begin{equation}\n\\label{intV_equ}\nV(T)=-g \\int_{m(T_0)}^{m(T)}  dm\n \\int \\frac{d^3p}{(2 \\pi)^3} \\frac{m}{E(p)}\nf(E(p)) +V(T_0)\\  ,\n\\end{equation}\nwhere $g=\\sum_i g_i$.\n\nWe apply the above methods to the lattice data from \\cite{lattice}\nshown in Fig. \\ref{lat_fig}.  These data\n are calculated for 4 flavors, corrected\nfor finite lattice size and extrapolated to massless fermions.\n We use  $g=62.8$ corresponding to the noninteracting limit of\n QCD with $4$ massless flavors.\nWe  draw the attention to the fact that the lattice data in\nFig. \\ref{lat_fig}\nfor  $\\epsilon_{lat}(T)\/T^4$  do not approach\nthe Stefan-Boltzmann limit for $T \\gg T_c$.\nThis may have two reasons.\n\\begin{itemize}\n\\item [(a)] The parton mass $m(T)$ never approaches the chiral limit\n$m=0$. Perturbative arguments (whose validity is questionable around\n$T_c$) suggest that $m(T)\\sim T$ for large $T$. If we attribute the\ndiscrepancy between $\\epsilon_{lat}(T)\/T^4$ and the Stefan-Boltzmann\nlimit to this reason, one finds the minimal mass $m(T)\\simeq 2.1 T_c$\nright above $T_c$ and $m(T)\\simeq 1.1 T$ for large $T$.\n\\item [(b)] Chiral restoration requires\n that $m(T)\\rightarrow 0$  above the phase\ntransition at least for the fermions. Then  the deviation from the\nStefan-Boltzmann limit has to be attributed \nto a mechanism which is outside the scope of the\nquasi-particle approach.\n\\end{itemize}\n\\begin{figure}[] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{f2.eps}\n\\caption[]{The dashed line represents the \ntemperature dependence of the quasiparticle mass \nas obtained from the lattice data (case (a)).\nThe short dashed line represents the extrapolation to the low temperature\nregion. It is taken to be constant, and suffices to effectively confine\nthe partons (an absolute confinement would require that the mass goes\nto infinity at low temperatures). The  solid line represents the \nassumed temperature dependence of the parton mass with zero limit at\nhigh temperature (case (b)).}\n\\label{mass_fig}\n\\end{center} \n\\end{figure}\nWe will explore both possibilities and call the corresponding masses\n$m_a(T)$ and $m_b(T)$ respectively.\nFirst we solve the Eq. (\\ref{diff_equ}) to obtain $m_a(T)$ for the case\n(a),\nwhere we set $m_a(T_0)=9.5T_c$ for $T_0=0.8T_c$.\nFor the case (b) we take $m_a(T)$ obtained in (a) for $T < T_c$ and\nextrapolate it at $T>T_c$ so that $m(T) \\rightarrow 0$ at large $T$\n(Fig. \\ref{mass_fig}).\nFrom this new functional form $m_b(T)$ we obtain $V_b(T)$ and\n$\\epsilon_b(T)$ via Eq. (\\ref{intV_equ}).\nFig. \\ref{mass_fig} shows the obtained temperature dependence of the parton mass for\nthe two cases. While the two functions $m(T)$ coincide\nin the region of the confinement transition (below $T_c$) they differ\ndramatically for \n$T>T_c$. The mass $m_a(T)$ increases linearly with $T$\nfor large temperatures while $m_b(T)$ is set to reach the chiral limit\nat high temperatures.\n Fig. \\ref{lat_fig}  also shows also the energy density\ncorresponding to the two cases. We observe that in the case (a) \n $\\epsilon_{qp}(T)$ follows  the lattice points as it should.\n In particular it has the same high temperature\nlimit which is different from the Stefan-Boltzmann limit.\n\\begin{figure}[t] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{f3.eps}\n\\caption[]{The potential energy density $V(T)$ as a function of the\ntemperature for the case (a) and (b) of the temperature dependence of\nthe mass, dashed and solid line respectively.}\n\\label{pot_fig}\n\\end{center} \n\\end{figure}\n\\noindent The energy density for the case (b) is different for $T>T_c$ and\napproaches the Stefan-Boltzmann limit. \n Fig. \\ref{lat_fig}  also shows the densities for the  kinetic energy\n $\\epsilon_{kin}(m(T),T)$ and for the potential energy $V(T)$\n for the case (b).\nFig. \\ref{pot_fig}\n shows the potential density functions $V(T)$ for both  cases. \nIn the case (b), one can identify the large temperature\nlimit of $V_b(T)$ with the bag constant $B$, since $V_b(T=0)=0$.\n Indeed for large $T$ (in the\ndeconfined phase) Eqs. (\\ref{press_equ}) and (\\ref{ener_equ}) take the\nform of the free massless gas with a bag constant\n$B=V_b(T=\\infty)$. \nFor the case (a) one can {\\it define} the bag constant as the\nvalue of the potential density $V_a(T)$ for the temperature where the mass\n $m_a(T)$ is the smallest (slightly above $T_c$). \nWe find\n\\begin{equation}\nB^{1\/4} =\\left\\{ \\begin{array}{rcll} \n1.35T_c & = & 240 \\ \\textrm{MeV} & \\ \\ \\textrm{ for case (a)} \\\\\n1.67T_c & = & 300 \\ \\textrm{MeV} & \\ \\ \\textrm{ for case (b)} \n\\end{array} \\right.\n\\end{equation}\nfor a value of $T_c=180$ MeV corresponding to four-flavor QCD.\n The order of magnitude of the bag constant is correct, but in\norder to compare it to the usual bag constant ($B^{1\/4}=135-200$ MeV)\n one should use the lattice\nresults for the energy density and\nthe critical temperature of the two-flavor QCD as an input for the \nquasi-particle parton model.\n\nIn order to distinguish between the two solutions for $m(T)$ we\ninvestigate the temperature dependence of the chiral condensate\nin the quasi-particle picture\n\\begin{equation}\n\\label{chi_equ}\n\\langle {\\bar \\Psi} \\Psi  \\rangle_{qp} (T) = \n\\langle {\\bar \\Psi} \\Psi  \\rangle_{vac} (T) + g_{q{\\bar q}}\n\\int \\frac{d^3p}{(2 \\pi)^3}\\frac{m}{E} f(E) \\ ,\n\\end{equation} \nwhere $g_{q{\\bar q}}$ counts  fermion and anti-fermion\n degrees of freedom \nand the vacuum part of the chiral condensate is given by an expression\nincluding a cutoff in momentum\n\\begin{equation}\n\\langle {\\bar \\Psi} \\Psi  \\rangle_{vac} (T) =  g_{q{\\bar q}}\n\\int_{|p|<\\Lambda} \\frac{d^3p}{2(2 \\pi)^3}\\frac{m}{E}  \\ .\n\\end{equation} \n\\begin{figure}[b] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{f4.eps}\n\\caption[]{The  chiral condensate as a function of the temperature\n for the case (a) and (b) of the temperature dependence of the mass,\n dashed and solid line respectively. The data points are for\nthe lattice data in the four-flavor QCD \\cite{kar}.}\n\\label{ch_fig}\n\\end{center} \n\\end{figure}\n\\noindent\nThe value of the cutoff $\\Lambda=430$ MeV is fixed to reproduce the zero\ntemperature value of the chiral condensate which we take \ntwice the usual value\n\\begin{equation}\n\\langle {\\bar \\Psi} \\Psi  \\rangle_{vac} (0) =  2 (250 \\textrm{MeV})^4 \\ ,\n\\end{equation} \nbecause we are modeling the  four-flavor lattice QCD.\nFig. \\ref{ch_fig} shows a comparison between the condensate functions for the\nquasi-particle picture and the lattice data. This comparison clearly\nfavors case (b), where $m(T)\\rightarrow 0$ above $T_c$. In\nwhat follows, we will mainly work with this solution.\n\n\n\nBefore we treat the nonequilibrium case\nwe apply the gap  equation to the case of finite density  and \ncalculate \nthe density dependence of the parton mass.\nWe  introduce variables which will be also useful for the\ndiscussion of the nonequilibrium case. \nFirst let us write the gap equation in the form\n\\begin{equation}\n\\frac{dV}{dm}=V'(\\rho)\\frac{d\\rho}{dm}=-\\rho \\ ,\n\\end{equation}\nwhere we define the scalar density\n\\begin{equation}\n\\rho=\n g \\int \\frac{d^3p}{(2 \\pi)^3} \\frac{m}{E(p)}\nf(E(p)) \\ .\n\\end{equation}\n\\begin{figure}[b] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{f5.eps}\n\\caption[]{The  quasiparticle mass  as a function of the scaled \nFermi momentum $p_f\/T_c$. The solid and  dashed lines correspond to two\ndifferent temperature dependences of the parton mass in Fig. 2.\n}\n\\label{fd_fig}\n\\end{center} \n\\end{figure}\n\\noindent\nIt allows us to use the functions $V(\\rho)$ and $m(\\rho)$ instead of $V(T)$\nand $m(T)$ for the parameterization of the potential energy density and\nthe mass respectively.\nAssuming that the energy density depends on temperature $T$ and chemical\npotential $\\mu$ only through  $\\rho(T,\\mu)$ we can generalize\nthe finite temperature case also to finite density. In principle the\npotential $V(\\rho)$ could depend on  other quantities,  e.g. the\nbaryon density. Any such more general case cannot be discussed using\nonly the lattice data at finite temperature.\nAs a support for our assumption we note that for the\nNJL model the potential density $V$  depends only \non the density $\\rho$ (Sec. \\ref{other_sec}).\n\nWe  write the gap equation (\\ref{gap_equ}) at finite density\n\\begin{equation}\n\\label{gapden_equ}\n\\frac{dV}{dm}=- g_f \\int \\frac{d^3p}{(2 \\pi)^3} \\frac{m}{E(p)}\n\\Theta(p_f-|p|) \\ ,\n\\end{equation}\nwhere $g_f$ counts the number of fermion degrees of freedom.\nNote that here we are using the Fermi distribution for the fermions\nat finite density and zero temperature and $V(m)$ from the lattice data.\nFig. \\ref{fd_fig}\n shows the mass of the partons as a function of the Fermi momentum $p_f$.\nThe behavior is similar as in the finite temperature case.\nFor low density the mass increases leading to the effective confinement.\nAt high density the mass is proportional to $p_f$ for the case (a) and\ngoes to zero in the case (b).\n\n\n\n\n\n\\section{Transport theory of the effective confining model}\n\n\n\nIn this section we will discuss the nonequilibrium evolution of the \nparton densities. In the semiclassical limit the collisonless \n plasma of quasi-particle partons\n can be described by the  Vlasov equation for the phase-space\ndistribution function $f(x,t,p)$:\n\\begin{eqnarray}\n\\label{vlasov_equ}\n\\partial_t f(x,t,p)+\\frac{ p}{E(p,x,t)} \\nabla_x f(x,t,p)\n& & \\nonumber \\\\\n- \\frac{m(x,t)}{E(p,x,t)} \\nabla_x m(x,t) \\nabla_p f(x,t,p)=0 \\ .\n\\end{eqnarray}\nEq. (\\ref{vlasov_equ}) has to be supplemented by an equation for the\nspace-time dependent mass  $m(x,t)$ .\nA sufficient condition for the requirement, that the Vlasov equation\ndescribes the same physics at thermal equilibrium as presented\nin the\nprevious section is that $m(x,t)$ satisfies the same gap equation  \n\\begin{equation}\n\\label{gapgen_equ}\n \\frac{dV}{dm} = -  g  \\int \\frac{d^3p}{(2 \\pi)^3}\n\\frac{m(x,t)}{E(p,x,t)} f(x,t,p) = -\\rho(x,t) \\ ,\n\\end{equation}\nwhere the thermal distribution function $f(E)$  in Eq. (\\ref{gap_equ})\nhas been replaced by the nonequilibrium solution of the Vlasov\nequation $f(x,t,p)$ and where $V(m)$ is the same as in equilibrium.\n The solution of the nonequilibrium gap equation is \nthen a function $m(x,t)$ of space and time.\nEq. (\\ref{gapgen_equ}) is however not the most general equation which reduces\nto the equilibrium finite temperature gap Eq. (\\ref{gap_equ}),\nbut one can add to it  terms which depend on the\nspatial and time derivatives of $m(x,t)$.\nWe will come back to this question in Sec. \\ref{other_sec}.\nHere we note that the choice in Eq. (\\ref{gapgen_equ})\n guarantees that the total energy of the system\n\\begin{eqnarray}\nE(t)& =& g \\int d^3x \\int \\frac{d^3p}{(2 \\pi)^3} E(p,x,t)\nf(x,t,p) \\nonumber \\\\\n & & +\\int d^3x \/ V(\\rho(x,t)) \n\\end{eqnarray}\nis  conserved by the evolution according to the Vlasov\nEq. (\\ref{vlasov_equ}), i.e. the energy of the system is constant.\n\nWe have numerically solved  the Vlasov equation together\nwith the gap equation for the two cases of the functional dependence of\nthe mass $m(T)$ on the temperature discussed in Sec. \\ref{sec_2}, but \nmost of the results shown relate to the case where the chiral condensate\nvanishes at high temperatures (case (b) in the previous section).\n We have used the test particle method for the solution of the Vlasov\nequation, i.e. we have made the ansatz\n\\begin{equation}\nf(x,t,p)=\\sum_{j=1}^{N}\\delta^3(x-x_j(t))\\delta^3(p-p_j(t)) \\ ,\n\\end{equation}\nwhere the trajectories $x_j(t)$ and $p_j(t)$ of the $N$ test particles\n satisfy Hamilton's\nequations with\n\\begin{equation}\nH(x,p)=\\sqrt{ p^2+m^2(x,t)} \\ .\n\\end{equation}\nThe initial conditions $x_j(0)$ and $p_j(0)$ are chosen so that\ngiven  initial densities for  matter and momentum  are reproduced.\nThe initial density is chosen spherically symmetric. Also in the \nsolution of the\n gap equation  the spherical symmetry is imposed by angle averaging. \n\n\\begin{figure}[b] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{density.ps.eps}\n\\caption[]{The  parton density distribution \n at different times,\nas obtained from the nonequilibrium evolution of the \ninitial fireball ($t=0$).}\n\\label{den_fig}\n\\end{center} \n\\end{figure}\nAt time $t=0$ the system is described by the density profile shown in\nFig. \\ref{den_fig} and with a momentum distribution corresponding to a\ntemperature $T=1.3T_c=180$ MeV. Fig. \\ref{den_fig} shows the density for\ndifferent times. The system expands for $t=2$ and $6$ fm\/c, but then comes\nback ($t=10$ fm\/c). The dependence of $m(x,t)$ on the radius at different\ntimes is shown in\nFig. \\ref{masstime_fig}. As expected\n\\begin{figure}[b] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{mass.ps.eps}\n\\caption[]{The  parton mass distribution at different times,\nas obtained from the nonequilibrium evolution of the\nfireball.}\n\\label{masstime_fig}\n\\end{center} \n\\end{figure}\n the mass is small in the interior and\nincreases towards the surface. This increase is responsible for the\nconfinement.   Indeed one can show that the equation\n of motion of one particle in the mean-field of the other particles\nconserves the  energy of the particle $\\sqrt{p^2+m^2}$. Thus\na particle cannot leave the region of deconfined plasma, if\nits initial momentum $p$ satisfies:\n\\begin{equation}\n\\label{conf_equ}\np^2 < m_{vac}^2-m^2 \\ ,\n\\end{equation}\nwhere $m_{vac}$ is the parton mass in vacuum and $m$ is the initial \nselfconsistent mass of the parton inside the plasma. The vacuum mass\nshould be \ninfinite, if the confinement is absolute. In our calculation we take \nthe vacuum mass equal to $\\sim 9.4 T_c$\n (see Fig. \\ref{mass_fig}), which effectively confines the\n partons at the temperatures discussed here. \n Thus the partons cannot leave the hot fireball if their initial \nmomentum is smaller than $\\sim 9.4 T_c$,\n which is the case for most of the partons \nat our initial temperature. \n\n Figs. \\ref{px_fig} and \\ref{x_fig}  show \nthe time development of the momentum $p_x$ and the coordinate $x$ of\na particular test particle.\n The particle oscillates between \nthe borders of the fireball and at the border its momentum is reduced and \neventually reversed by the action of the force:\n\\begin{equation}\n\\frac{dp}{dt}=-\\frac{m}{E}\\nabla_x m \\ .\n\\end{equation}\n\nIn contrast to the simple particle motion, partons traveling in bunches\nmay leave into the vacuum region, since\n\\begin{figure}[t] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{trajec_px.eps}\n\\caption[]{The component $p_x$ of the momentum of a \nparticular test-particle\n taken from the\nsimulation of the time evolution of the region of deconfined plasma.}\n\\label{px_fig}\n\\end{center} \n\\end{figure}\n\\begin{figure}[h] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{trajec_x.eps}\n\\caption[]{The  $x$ coordinate of the same test-particle\nas in Fig. \\ref{px_fig}.}\n\\label{x_fig}\n\\end{center} \n\\end{figure}\n the internally created field leads\nto small masses inside the bunch. This possibility is excluded in our\nmethod of \n\\noindent\nsolution, since we require at each time step that the system stays \n  spherically symmetric. This mechanism then\n generates     collective vibrations of the surface of the\n fireball, when particles are trying to leave the hot region \n\\begin{figure}[b] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{r2mean.eps}\n\\caption[]{The time evolution of the mean square radius of the region of\ndeconfined plasma for the case (a) and (b), dashed and solid  line\nrespectively}\n\\label{r2_fig}\n\\end{center} \n\\end{figure}\n\\noindent simultaneously.\n Fig. \\ref{r2_fig}  shows the time dependence for the mean\n square  radius of the fireball~:\n\\begin{equation}\n\\langle r^2(t) \\rangle =  g\\int d^3x \\ x^2   \\int \\frac{d^3p}{(2 \\pi)^3} \nf(x,t,p) \\ .\n\\end{equation}\nThe oscillation which can be seen on the figure reflect the collective \nmonopole oscillations of the parton density, with  partons  leaving\nthe fireball and   reflected back when their mass grows.\nAnalogous collective oscillations have been observed in the Vlasov\nevolution of the nucleon in the Friedberg-Lee model \\cite{giessen}.\n\nAs the volume of the\n fireball  oscillates,  its potential energy will also oscillate \n growing for large volumes and decreasing when the system is compressed.\nThe period of the collective oscillations is basically determined \nby the time which a particle needs to travel from one border of the\nfireball to the other (Fig. \\ref{x_fig}). For  massless\ndeconfined partons (case (b) in Sec. \\ref{sec_2}) this time is\ntwice the radius of the system divided by the speed of light.\nThe total energy is of course conserved (see Fig. \\ref{ener_fig}),\n to the accuracy of \nour numerical solution, and the kinetic (and then also the potential energy)\n has oscillations of the same period  as the oscillations of the \nmean square radius in Fig. \\ref{r2_fig}.\n\nWe mention the  possibility of  fragmentation\n of the fireball into smaller pieces, each of them having \nlarge parton density inside and thus a \nsmall mass of partons. This fragmentation\nmechanism cannot be studied in our spherically symmetric mean-field theory.\nIt would require a description including the fluctuations of the density.\n\n\\begin{figure}[b] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{energy.eps}\n\\caption[]{The time dependence of the total and potential energy \nof the parton plasma, solid and  dashed lines respectively}\n\\label{ener_fig}\n\\end{center} \n\\end{figure}\n\nWe state the main result of this section: the Vlasov equation together\nwith the gap equation based on lattice data shows confinement.\nFor not too high initial temperatures the fireball stays compact. \nOf course  the surface of the fireball\n can oscillate, but the system remains bounded. This\nbehavior is in contrast to the free expansion of the parton gas, where the\nparton fireball would start to expand and cool down faster.\nIn our picture the partons are confined, since no hadronization is included\n(see Fig. \\ref{r2_fig}).\n\n\n\\section{Relation to other effective models}\n\\label{other_sec}\n\nThe effective models of the QCD using parton degrees of freedom\n  are mostly restricted to the fermion\n sector. Thus NJL type models \\cite{NJL} have only fermionic degrees of\n freedom with four-fermion interaction. At high temperature and\/or\n density the quarks are massless or almost and at low densities due to\n the nonzero quark-condensate they acquire a finite mass with value\n around $350$ MeV. The quarks are not confined in this theory.\n The NJL gap equation\n for the quark mass can also be written in the form of \nEq. (\\ref{gap_equ}) where $dV\/dm$ is defined by the expression\n\\begin{equation}\n\\frac{dV}{dm} \\equiv \\frac{(m-m_0)}{2G} - 6 N_f\n\\int_{|p|<\\Lambda} \\frac{d^3p}{(2\\pi)^3} \\frac{m}{E} \\ ,\n\\end{equation}\nwhere $G$ is the four fermion coupling constant, $\\Lambda$ is the\ninfrared cutoff, $m_0$ is the current quark mass and $N_f$ is the number\nof flavors. Fig. \\ref{pote_fig} shows a comparison between the potential\nextracted from lattice calculations and the one from the NJL model.\nThe difference is twofold: \n(i) Shape: While the potentials for Friedberg-Lee and NJL model show a\nclear minimum at the position of the constituent mass in the vacuum, the\npotential from our approach drops monotonically to zero, reflecting the\nconfinement (infinite vacuum mass).\n(ii) Magnitude: At $m=0$ the potential from our approach differs by about\na factor $5$ from the other approaches. This may be partly due to the\ndifferent numbers of degrees of freedom. While Friedberg-Lee and NJL\nrefer to a two-flavor quark theory (no gluons), the lattice calculation\nis performed for $N_f=4$ and gluons.\nThe remaining difference may be due to quantitative difference between\nthe four-flavor and the two-flavor QCD, which goes beyond a simple\nrescaling of the number of degrees of freedom.\n  \n\n\\begin{figure}[b] \n\\begin{center}\n\\includegraphics[width=9cm,angle=0]{veff.eps}\n\\caption[]{The potential energy density $V(m)$ as a function of the mass\nas extracted from lattice QCD \n(solid line), for the two-flavor NJL model (dashed line)\nand for the Friedberg-Lee model (dotted line).\nWe used $T_c=140$ MeV  in defining $V(m)$.\nThe large value of the bag constant $B=V(m=0)$ for the potential density \nfrom the lattice data can be due to a larger number of degrees of freedom\nin the four-flavor model. }\n\\label{pote_fig}\n\\end{center} \n\\end{figure}\n\n\nThe Friedberg-Lee model describes fermions coupled to a scalar field\n$\\sigma(x,t)$ which plays the role of an effective mass and whose\ndynamics is driven by a potential $U(\\sigma)$.\nOf course as in the NJL model the degrees of freedom are restricted to\nthe fermions. The Lagrangian of the Friedberg-Lee model can be written as:\n\\begin{equation}\n\\label{fl_equ}\n{\\cal L}={\\overline \\Psi}(i\\gamma^\\mu\\partial_\\mu)\\Psi -\ng \\sigma {\\overline \\Psi}\\Psi +\\frac{1}{2}\\partial_\\mu\\sigma\\partial^\\mu\n\\sigma-U(\\sigma) \\ ,\n\\end{equation}\nwhere the fermion field operator $\\Psi$ also carries the flavor and color\nindices. \nThe effective quark mass $g \\sigma$ can include also a contribution from\nthe current quark mass. \n\nWhen the expectation value of the $\\sigma $ \nfield is infinity, the quarks are confined. The same is  also\neffectively true\nif the vacuum expectation value of the $\\sigma$ field is very large.\nThe gap equation for the quark mass $\\sigma$ is given by the classical\nequation of motion for the $\\sigma$ field:\n\\begin{equation}\n\\Box \\sigma + U^{'}(\\sigma)= - g < {\\bar \\Psi}\\Psi > (x,t) \\ .\n\\end{equation}\n\nIn the case of homogeneous systems $\\Box \\sigma=0$ (e.g. in the\nmean-field thermodynamics)\nthe Friedberg-Lee gap equation is equivalent to the gap equation used in\nSec. \\ref{sec_2} if $V(m)=U(\\sigma)$.\nIn particular the thermodynamical energy density discussed in\nSec. \\ref{sec_2} can be reproduced in the Friedberg-Lee model \nif not for the neglect of the gluon degrees of freedom. The difference\nbetween our approach and the one by Friedberg and Lee rests in the\nchoice of the potential. While they assume certain forms,\nwe let $V(m)$ be determined by the lattice data.\nOne should note that in the homogeneous systems the confining\nFriedberg-Lee model, with the fermion mass $m=\\kappa(\\sigma)$\nbeing a function of the field $\\sigma$, is also equivalent to our\napproach after a change of variables $\\sigma \\rightarrow m$.\nThe infinite value of the fermion mass  means simply in\nthe language of Eq. (\\ref{fl_equ}), that the \nvacuum gap equation $dV\/dm=0$ has a solution at $m=\\infty$.\n\n The kinetic term \nfor the $\\sigma $  makes a difference for the case of nonequilibrium \nor nonhomogeneous systems. In the dynamical evolution of a\nnonequilibrium system in the Friedberg-Lee model, the $\\sigma $ field\nis another dynamical field not related to the local value of the\nscalar density $\\rho$ \\cite{giessen}. The inclusion of the kinetic term\nfor the $\\sigma $ field however leads to the problem that the value of\nthe $\\sigma$ field can go negative. The potential $U(\\sigma)$ cannot be\nextracted for the negative values of $\\sigma$ from the lattice\ndata. Moreover in the cases when the mass of the fermion field becomes\nnegative, its evolution cannot be described by a semiclassical Vlasov\nequation.\n\n\n\\section{Discussion}\n\n\nThe description of the deconfinement transition in heavy ion\ncollisions is  a very important theoretical problem. A realistic\ndescription could allow to define which observables are relevant \nfor the observation of the quark-gluon plasma formation. A\ndynamical simulation is wished for in order to extract the properties of\nthe plasma from the experimental data. Any such approach\n meets difficulties in the description of the confinement and of \nthe hadronization\ntransition when the temperature of the deconfined region drops down.\nIn the present work we have addressed only a part of this program,\nnamely the influence of the confinement on the dynamics of partons.\n\nThe confinement of partons below $T_c$ is described in a quasi-particle\ngas model by the increase of the parton mass at low energy densities.\n The temperature dependence\nof the parton mass is extracted from the lattice data. \nIn the  range\nof temperatures analyzed ($0.8T_c <T< T_c$) the mass of the partons\nincreases to values which effectively confine the quarks and gluons.\n  The requirement of the restoration of the chiral symmetry at high \ntemperatures fixes the temperature mass dependence for $T>T_c$.\n\nThe formalism is generalized to the nonequilibrium case.\nThe time development of a deconfined\nfireball  studied in a Vlasov equation \nis different from what is  usually discussed in the literature,\nin that it shows confinement. The expansion of the system\nis forbidden, but instead\nwe observe collective oscillations of the parton plasma.  The time scale\nof these oscillations in the collisonless plasma is determined by the\nsize of the deconfined region. This time scale should be compared to the\nhadronization time in order to determine if the oscillation could\ndevelop.\n\nThe increase of the parton mass has implication also for the \nhydrodynamical model of the plasma expansion.\nThe slowing down of the hydrodynamic expansion is observed  in\n simulations of systems with first order phase transition\n\\cite{sh} or in a hydrodynamical calculation with the NJL model \\cite{copen} .\nThe use of a confining mass in the hydrodynamical model would \nstop the expansion.\nFurther expansion of the fireball is possible only after\nhadronization. \n\nIf the hadronization takes place mainly  \nat the surface of the deconfined phase\nthen the dynamics of the system could be described by \na hydrodynamical model with\nfirst order phase transition. However, the description using the\ntransport equation allows to discuss alternative scenarios of\n the hadronization, e.g.  hadronization due to parton collisions in the\nplasma \\cite{NJL_had} with possible softening of the spectra of produced \nmesons. Another hadronization mechanism could be the fragmentation of\nthe fireball due to a possible instability of the system at finite\nbaryon density \\cite{ins} or due to dynamical instabilities present for energy\ndensities corresponding to a mixed phase in the case of a first order\nphase transition. The discussion of the hadronization mechanism and the\ninclusion of the dynamics of mesons remains to be done.\n\n\n\n\n\\section*{Acknowledgments}\nOne of the authors (P.B.) wishes to thank the Alexander von Humboldt\nFoundation for financial support. This work has been supported\nin part by the German Ministry for Education and Research (BMBF)\nunder contract number 06 HD 742.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\nThe classification of closed smooth $d$-manifolds and families thereof---smooth fibre bundles---is one of the guiding problems of geometric topology. From a homotopy-theoretic perspective, it is the study of the $\\infty$-groupoid\\footnotemark[1] $\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}$ of smooth closed $d$-manifolds and spaces of diffeomorphisms between them. A historically successful approach to relate---and partially reduce---the study of $\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}$ in high dimensions to more homotopy-theoretic and algebraic questions goes by comparison to the $\\infty$-groupoid $\\ensuremath{\\catsingle{S}}^{\\simeq}$ of spaces via the functor $\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}\\rightarrow \\ensuremath{\\catsingle{S}}^{\\simeq}$ that assigns a manifold its homotopy type. For a given homotopy type $X$, one studies the fibre\n\\[\n\tS^{\\ensuremath{\\catsingle{S}}}(X)\\coloneqq \\mathrm{fib}_X\\big(\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong} \\rightarrow \\ensuremath{\\catsingle{S}}^{\\simeq}\\big),\n\\]\nwhich can be thought of as the space of \\emph{manifold structures} on $X$. The path components of this \\emph{structure space} are equivalence classes of manifolds with a homotopy equivalence to $X$,\n\\[\n\t\\pi_0\\,S^{\\ensuremath{\\catsingle{S}}}(X) = \\frac{\\left\\{\\text{\\parbox{7cm}{\\centering pairs $(M,\\varphi)$ of a closed smooth $d$-manifold $M$ and \\newline a homotopy equivalence $\\varphi \\colon M \\to X$}}\\right\\}}{\\parbox{7.5cm}{\\centering $(M,\\varphi) \\sim (M',\\varphi')\\Leftrightarrow$ there exists a diffeomorphism $\\alpha \\colon M \\to M'$ with $[\\varphi' \\circ \\alpha]=[\\varphi]\\in\\pi_0\\,\\mathrm{Map}_\\ensuremath{\\catsingle{S}}(M,X),$}}\n\\] \nand the path component of $S^{\\ensuremath{\\catsingle{S}}}(X)$ corresponding to such a pair $(M,\\varphi)$ agrees with the identity component of the fibre $\\ensuremath{\\mathrm{hAut}}(M)\/\\ensuremath{\\mathrm{Diff}}(M)$ of the map $\\ensuremath{\\mathrm{BDiff}}(M)\\rightarrow\\ensuremath{\\mathrm{BhAut}}(M)$ induced by considering diffeomorphisms as homotopy equivalences,\n\\[\n\tS^{\\ensuremath{\\catsingle{S}}}(X)_{(M,\\phi)} \\simeq \\big(\\ensuremath{\\mathrm{hAut}}(M)\/\\ensuremath{\\mathrm{Diff}}(M)\\big)_{\\mathrm{id}}.\n\\]\nSurgery theory and pseudoisotopy theory combine to provide an approximation to the structure space $S^{\\ensuremath{\\catsingle{S}}}(X)$ up to extensions in terms of three infinite loop spaces---one in the realm of each, \\emph{algebraic $K$-theory},  \\emph{algebraic $L$-theory}, and \\emph{stable homotopy theory} (see \\cite{WWsurvey} for a survey). The unfortunate defect of this approach is that it really is only an approximation, in the sense that it can only capture a finite Postnikov truncation of $S^{\\ensuremath{\\catsingle{S}}}(X)$ depending on the dimension.\n\n\\footnotetext[1]{This work is written $\\infty$-categorically, so we treat homotopy types and $\\infty$-groupoids as indistinguishable. In this introduction, readers unfamiliar with this principle may substitute topologically enriched categories or groupoids for $\\infty$-categories or -groupoids; the former being related to homotopy types by taking classifying spaces.}\n\n\n\\medskip\n\nMotivated by Goodwillie--Weiss' embedding calculus and factorisation homology, we pursue a different approach to relate the study of $\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}$ to more homotopy-theoretic and algebraic questions, and we establish three fundamental properties of this alternative. Observing that the homotopy type of a manifold $M$ can be viewed as that of the space of ordered configurations of $k$ points in $M$ for $k=1$, this approach is motivated by the idea to remember the homotopy types of the configuration spaces for \\emph{all} values of $k$, together with the natural point-forgetting maps between them. It is in fact beneficial to consider configuration spaces of thickened points which admit more natural maps between them, by ``splitting points''. To make this precise, one considers the $\\infty$-category $\\ensuremath{\\icat{D}\\mathrm{isc}}_d$ of finite disjoint unions of $d$-dimensional Euclidean spaces, i.e.~$T\\times \\ensuremath{\\mathbf{R}}^d$ for finite sets $T$, and spaces of smooth embeddings between them. A $d$-manifold $M$ gives rise to a presheaf $E_M\\colon \\ensuremath{\\icat{D}\\mathrm{isc}}_d^{\\mathrm{op}}\\rightarrow \\ensuremath{\\catsingle{S}}$ on $\\ensuremath{\\icat{D}\\mathrm{isc}}_d$ with values in the $\\infty$-category $\\ensuremath{\\catsingle{S}}$ of spaces via\n\\begin{equation}\\label{equ:presheaf-dfn}\n\t\\ensuremath{\\icat{D}\\mathrm{isc}}_d^\\mathrm{op}\\ni T\\times \\ensuremath{\\mathbf{R}}^d\\xmapsto{E_M}\\ensuremath{\\mathrm{Emb}}(T\\times \\ensuremath{\\mathbf{R}}^d,M)\\in\\ensuremath{\\catsingle{S}}.\n\\end{equation}\nBy taking derivatives at the centres, the space $E_M(T\\times \\ensuremath{\\mathbf{R}}^d)$ is equivalent to the ordered configuration space of $k=|T|$ points in $M$ together with framings of the tangent space of $M$ at each of these points, and the homotopy type of the ordinary ordered configuration space of $k$ points in $M$ (in particular that of $M$ itself for $k=1$) can be recovered as the quotient by the $\\ensuremath{\\mathrm{Diff}}(\\ensuremath{\\mathbf{R}}^d)^{T} \\simeq \\mathrm{O}(d)^T$-action on $E_M(T\\times \\ensuremath{\\mathbf{R}}^d)$ obtained by functoriality. The assignment $M\\mapsto E_M$ as in \\eqref{equ:presheaf-dfn} is natural in embeddings of $M$, so it in particular defines a functor $E\\colon \\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}\\rightarrow \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)^\\simeq$ to the $\\infty$-groupoid of $\\ensuremath{\\catsingle{S}}$-valued presheaves on $\\ensuremath{\\icat{D}\\mathrm{isc}}_d$. The fibre of this functor at a presheaf $X \\colon \\ensuremath{\\icat{D}\\mathrm{isc}}^{\\mathrm{op}}\\rightarrow \\ensuremath{\\catsingle{S}}$\n\\begin{equation*}\\label{equ:def-disc-structure-space}\n\tS^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(X)\\coloneqq \\mathrm{fib}_X\\big(\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}\\xra{E} \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)^\\simeq\\big)\n\\end{equation*}\nis the eponymous \\emph{$\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space of $X$}. Analogous to the more traditional structure space $S^{\\ensuremath{\\catsingle{S}}}(X)$, the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(X)$ can be thought as a space of manifold structures, this time on a presheaf as opposed to just a homotopy type. Similar to before, the path components $\\pi_0\\,S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(X)$ are represented by pairs of a manifold with an equivalence of its presheaf to $X$,\n\\[\n\t\\pi_0\\,S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(X) = \\frac{\\left\\{\\text{\\parbox{7cm}{\\centering pairs $(M,\\varphi)$ of a closed smooth $d$-manifold $M$ and \\newline an equivalence of presheaves $\\varphi \\colon E_M \\to X$}}\\right\\}}{\\parbox{9cm}{\\centering $(M,\\varphi) \\sim (M',\\varphi')\\Leftrightarrow$ there exists a diffeomorphism\\newline $\\alpha \\colon M \\to M'$ with $[\\varphi' \\circ E_\\alpha]=[\\varphi]\\in\\pi_0\\,\\mathrm{Map}_{\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)}(E_M,X)$,}}\n\\] \nand the path component of $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(X)$ corresponding to such a pair $(M,\\varphi)$ agrees with the identity component of the fibre $\\mathrm{Aut}(E_M)\/\\ensuremath{\\mathrm{Diff}}(M)$ of the map $\\ensuremath{\\mathrm{BDiff}}(M)\\rightarrow \\mathrm{BAut}(E_M)$ induced by $E$,\n\\[\n\tS^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(X)_{(M,\\varphi)} \\simeq \\big(\\mathrm{Aut}(E_M)\/\\ensuremath{\\mathrm{Diff}}(M)\\big)_{\\mathrm{id}}.\n\\]\nIn particular, the space $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(X)$ is nonempty if and only if $X\\simeq E_M$ for some closed smooth $d$-manifold $M$. If so, then $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(X)\\simeq S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(E_M)$ so nothing is lost by assuming $X=E_M$ in which case we abbreviate $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)\\coloneqq S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(E_M)$. These are the spaces we focus on in this work. Informally speaking, they measure by how many manifolds the presheaf $X=E_M$ is realised, and how much their diffeomorphism groups differ from the automorphism group of $X$.\n\n\\medskip\n\nAs the main results of this work, we establish three structural properties of $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ that one could summarise by saying that for most choices of $M$\n\\begin{enumerate}[A)]\n\t\\item $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ depends only little on the manifold $M$,\n\t\\item $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ is an infinite loop space, and\n\t\\item $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ is nontrivial.\n\\end{enumerate}\nWe state these results in terms of a more general version $S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ for manifolds that may have boundary, which is crucial for our methods. We postpone its definition to \\cref{sec:boundary-intro} below.\n\n\\renewcommand\\thesubsection{\\Alph{subsection})} \n\n\\subsection{Tangential $2$-type invariance}\nTo make the first property precise, recall that two manifolds $M$ and $N$, possibly with boundary, have the \\emph{same tangential $2$-type} if there is a map $B\\rightarrow \\mathrm{BO}$ so that the maps $M\\rightarrow \\mathrm{BO}$ and $N\\rightarrow\\mathrm{BO}$ classifying the stable tangent bundles of $M$ and $N$ admit lifts to maps $M\\rightarrow B$ and $N\\rightarrow B$ that are $2$-connected.\n\n\\begin{nex}\nChoosing $B=\\mathrm{BSpin} \\times K(\\pi,1)$, one sees that two spin manifolds $M$ and $N$ have the same tangential $2$-type if and only if their fundamental groupoids are equivalent. In particular, all simply connected spin manifolds have the same tangential $2$-type.\n\\end{nex} \n\nOur first main result is that in high dimensions, the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ depends only on the dimension $d$ and the tangential $2$-type of $M$.\n\n\\begin{bigthm}\\label{bigthm:2-type-invariance}\nFor compact $d$-manifolds $M$ and $N$ with $d\\ge5$ that have the same tangential $2$-type, there exists an equivalence $S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)\\simeq S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(N)$.\n\\end{bigthm}\n\nIn particular, the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space of a spin $d$-manifold $M$ with $d\\ge5$ only depends on the fundamental groupoid, so we in particular have $S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M) \\simeq S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(D^d)$ if $M$ is simply connected.\n\n\\begin{nrem}\nOne ingredient in the above mentioned approximation to the conventional structure space $S^{\\ensuremath{\\catsingle{S}}}_\\partial(M)$ has a similar invariance property (namely, the $L$-theory part depends only on the fundamental groupoid), but the others depend more substantially on the homotopy type of $M$.\n\\end{nrem}\n\n\\begin{nrem}\nReformulated in terms of embedding calculus (see \\cref{sec:emb-calc-intro2} for an outline of this relation), \\cref{bigthm:2-type-invariance} is an extension of a result of Knudsen--Kupers \\cite[6.23]{KnudsenKupers} which applies to certain path components of $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ if $M$ is $2$-connected, of dimension $d\\ge6$, and $\\partial M=S^{d-1}$.\n\\end{nrem}\n\n\n\\subsection{Infinite loop space structure}\nAs previously mentioned, the more traditional structure space $\\smash{S^{\\ensuremath{\\catsingle{S}}}_\\partial(M)}$ is an infinite loop space \\emph{after a certain truncation} and \\emph{up to extensions}. The $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ on the other hand is in high dimensions an actual infinite loop space---no truncations or extensions are necessary. This is our second main result.\n\n\\begin{bigthm}\\label{bigthm:infinite-loop-space}\nFor a compact $d$-manifold $M$ with $d\\ge8$, the space $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ admits the structure of an infinite loop space.\n\\end{bigthm}\n\n\\begin{nrem}\nThe bound $d\\ge8$ in \\cref{bigthm:infinite-loop-space} is not optimal. We show for example that it can be improved to $d\\ge6$ for simply connected spin manifolds (see \\cref{thm:oo-loop-general}). \n\\end{nrem}\n\n\\subsection{Nontriviality}\nAt this point a very optimistic reader may wonder whether the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ are just contractible, which would in particular say that the diffeomorphism group $\\ensuremath{\\mathrm{Diff}}(M)$ of a closed manifold $M$ is equivalent to the automorphism group $\\mathrm{Aut}(E_M)$ of the associated presheaf. As our third main result, we show that this is never the case as long as the manifold is assumed to be spin and of dimension $d\\ge5$.\n\n\\begin{bigthm}\\label{bigthm:nontrivial}\nFor a compact spin $d$-manifold $M\\neq\\varnothing$ with $d\\ge5$, the space $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ is not contractible.\n\\end{bigthm}\n\n\\begin{nrem}\nThere are partial results in low dimensions that complement \\cref{bigthm:nontrivial}.\n\\begin{enumerate}\n\t\\item For $d\\le 2$, Theorem A of \\cite{KrannichKupersSurfaces} implies $\\smash{S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial}(M)\\simeq \\ast$ (see Remark 1.1 (ii) loc.cit.).\n\t\\item For $d=3$, we give several examples for which $\\smash{S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial}(M)$ is nontrivial, including $M=D^3$ and $M=S^3$ (see \\cref{rem:3-manifolds}).\n\t\\item For $d=4$, Theorem B of \\cite{KnudsenKupers} implies that $\\pi_0\\,\\smash{S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial}(M)$ surjects onto the set of isotopy classes of smooth structures on $M$ as long as $M$ is $1$-connected and closed, so $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ is nontrivial for all such $M$ that admit more than one smooth structure.\n\\end{enumerate}\n\\end{nrem}\n\nThis concludes the summary of our three main results. In the remainder of this introduction, we briefly indicate how $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ relates to embedding calculus, the little $d$-discs operad, and factorisation homology, and then give a summary of the proofs of the main results, where we also make good for the omitted definition of $S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ for manifolds with boundary.\n\n\\renewcommand\\thesubsection{\\thesection.\\arabic{subsection}}\n\\setcounter{subsection}{0}\n\n\\subsection{Relation to embedding calculus, the $E_d$-operad, and factorisation homology}\n\n\\subsubsection{Embedding calculus}\\label{sec:emb-calc-intro2}\nGoodwillie and Weiss' \\emph{embedding calculus} \\cite{WeissImmersion,GoodwillieWeiss} is a device to study embeddings via their restrictions to submanifolds of the source that are diffeomorphic to $T\\times\\ensuremath{\\mathbf{R}}^d$ for finite sets $T$. It has the form of an approximation to the space of embeddings\n\\begin{equation}\\label{equ:emb-calc-intro}\n\t\\ensuremath{\\mathrm{Emb}}(W,W')\\longrightarrow T_\\infty\\ensuremath{\\mathrm{Emb}}(W,W')\n\\end{equation}\nwhose target is the limit of a tower of maps whose fibres admit a description in terms of the configurations spaces and frame bundles of $W$ and $W'$. The main result in this context, due to Goodwillie--Klein \\cite{GoodwillieKlein}, says that \\eqref{equ:emb-calc-intro} is an equivalence if the handle codimension (dimension of $W'$ minus handle dimension of $W$) is at least three. In general, the map \\eqref{equ:emb-calc-intro} can fail to be an equivalence, and in a sense the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces may be seen as the ``correction terms'' to \\eqref{equ:emb-calc-intro} being an equivalence in codimension zero. Let us make this more precise.\n\nThe relation of the map \\eqref{equ:emb-calc-intro} to $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces is a reformulation of a result of Boavida de Brito--Weiss \\cite{BdBWSheaf}, at least if $M$ is closed (c.f.\\,\\cref{rem:BdPW-other-boundary}). They show that \\eqref{equ:emb-calc-intro} is equivalent to the map $\\ensuremath{\\mathrm{Emb}}(W,W')\\rightarrow \\mathrm{Map}_{\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)}(E_W,E_{W'})$ induced by the naturality of $E_W$ in embeddings, which---for closed $W$ and $W'$ and after discarding non-invertible components in the target---is the map on mapping spaces induced by the functor $E\\colon \\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{\\cong}\\rightarrow \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)^{\\simeq}$ used to define the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space. This shows that the loop space of $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ based at $(N,\\varphi) \\in \\pi_0\\,S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ is equivalent to the fibre at $\\varphi$ of \\eqref{equ:emb-calc-intro} for $W=N$ and $W'=M$, so for $(M,\\mathrm{id})$ we get\n\\begin{equation}\\label{equ:rel-to-tinfty-looped}\n\t\\Omega S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)\\simeq \\mathrm{hofib}_{\\mathrm{id}}(\\ensuremath{\\mathrm{Emb}}(M,M)\\rightarrow T_\\infty\\ensuremath{\\mathrm{Emb}}(M,M)).\n\\end{equation}\n\n\\begin{rem}\\label{rem:BdPW-other-boundary}\nA similar discussion applies if $M$ has boundary, but this does not follow directly from \\cite{BdBWSheaf} since we deal with boundary conditions differently (see \\cref{sec:boundary-intro}). \n\\end{rem}\n\nSpecialising Properties A--C to spin manifolds, they in particular imply:\n\\begin{bigcor}\\label{cor:emb-calc-nontriviality}\nFor compact connected spin $d$-manifolds $M\\neq\\varnothing$ with $d\\ge5$, the fibre\n\\[\n\t\\mathrm{hofib}_{\\mathrm{id}}\\big(\\ensuremath{\\mathrm{Diff}}_\\partial(M)=\\ensuremath{\\mathrm{Emb}}_\\partial(M,M)\\rightarrow T_\\infty\\ensuremath{\\mathrm{Emb}}_\\partial(M,M)\\big)\n\\] \nis nontrivial and depends only on the fundamental group of $M$. It is an infinite loop space for $d\\ge8$.\n\\end{bigcor}\n\n\\subsubsection{The operad $E_d$ of little $d$-discs}\nWe continue by mentioning two connections between $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ and the operad $E_d$ of little $d$-discs. The first is that $\\ensuremath{\\icat{D}\\mathrm{isc}}_d$ agrees with the PROP associated to the framed $E_d$-operad, so $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ can be identified with the $\\infty$-category of right-modules over this operad and hence the definition of $S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ for closed manifolds can be rephrased in these terms. There is a similar reformulation if $M$ has boundary.\n\nThe second relation is less obvious and once more a result of work of Boavida de Brito and Weiss \\cite{BdBWConf}. To explain it, observe that the standard action of $\\ensuremath{\\mathrm{O}}(d)$ on the disc $D^d$ induces an $\\ensuremath{\\mathrm{O}}(d)$-action on the operad $E_d$ of little $d$-discs. This action extends to the topological group $\\mathrm{Top}(d)$ of homeomorphisms of $\\ensuremath{\\mathbf{R}}^d$, so there is a map\n\\begin{equation}\n\t\\label{equ:topd-to-ed-intro}\\mathrm{BTop}(d)\\longrightarrow \\mathrm{BAut}(E_d)\n\\end{equation}\nwith $\\mathrm{Aut}(E_d)$ the automorphism group of the $E_d$-operad; this can for instance be seen using a result of Boavida de Brito--Weiss' \\cite{BdBWConf}. Reformulated in our setting, their work moreover implies that there is an equivalence of the form\n\\begin{equation}\\label{equ:pedro-michael-equivalence}\n\t\\Omega^{d+2}(\\mathrm{Aut}(E_d)\/\\mathrm{Top}(d))\\simeq \\Omega S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_{\\partial}(D^d).\n\\end{equation}\nIn particular \\cref{bigthm:infinite-loop-space} and \\cref{bigthm:nontrivial} for $M=D^d$ (or rather certain refinements of them) imply:\n\n\\begin{bigcor}\\label{bigcor:top-vs-auted}\nThe map $\\mathrm{BTop}(d)\\rightarrow \\mathrm{BAut}(E_d)$ is an equivalence if and only if $d\\le2$. Moreover, its fibre admits for $d\\ge6$ the structure of an infinite loop space after taking $(d+2)$-fold loop spaces. \n\\end{bigcor}\n\n\\begin{rem}A couple of remarks on this corollary are in order.\n\t\\begin{enumerate}\n\t\t\\item Dwyer and Hess asked whether the map \\eqref{equ:topd-to-ed-intro} is an equivalence \\cite[58 min]{Dwyer}. The first part of \\cref{bigcor:top-vs-auted} gives an answer.\n\t\t\\item The cases $d\\le 2$ of the first part of \\cref{bigcor:top-vs-auted} are not due to us: Horel \\cite{Horel} proved the case $d=2$. The case $d=1$ is folklore and can be proved via Horel's approach.\n\t\\end{enumerate}\n\\end{rem}\n\n\\subsubsection{Factorisation homology}\\label{sec:factorisation-homology}\nThe final relation of $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ we would like to mention is one to \\emph{factorisation homology} (or \\emph{topological chiral homology}) \\cite{Salvatore, Francis, Andrade, AyalaFrancisTop, LurieHA}. In its simplest instance, this connection amounts to the (quite tautological) observation that for a framed $E_d$-algebra $A$ in a suitable $\\infty$-category $\\ensuremath{\\catsingle{C}}$, there is a commutative diagram \n\\[\\begin{tikzcd} \n\t\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}\\rar{E}\\arrow[dr,\"\\int_{(-)}A\",swap]&\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)\\dar{(-)\\otimes_{\\ensuremath{\\icat{D}\\mathrm{isc}}_{d}}A}\\\\\n\t&\\ensuremath{\\catsingle{C}}\n\\end{tikzcd}\\]\nof $\\infty$-categories in which the diagonal arrow is given by factorisation homology with coefficients in $A$ and the vertical arrow by taking coends, using that $A$ is in particular a functor $A\\colon \\ensuremath{\\icat{D}\\mathrm{isc}}_d\\rightarrow \\ensuremath{\\catsingle{C}}$. In fact, the functor $E$ itself is an instance of factorisation homology, namely with coefficients in the framed $E_d$-algebra $E_{D^d}\\in \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$, so $E$ may be viewed as the universal factorisation homology invariant on $\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}$, and the study of $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces as closely related to the question to which extent the theory of manifolds can be captured by factorisation homology.\n\n\\subsection{Summary of proofs}\nWe conclude with a summary of the proofs of Theorems~\\ref{bigthm:2-type-invariance}--\\ref{bigthm:nontrivial}. \\smallskip\n\n\\begin{center}\\textit{Some steps may be of independent interest. We highlight them with the Roman numerals \\ref{enum:bord-emb-calc}--\\ref{enum:rationalisation-operads-intro}.}\\end{center}\n\n\\subsubsection{The case with boundary}\\label{sec:boundary-intro}\nThe more general $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces $\\ensuremath{\\catsingle{S}}_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ for manifolds $M$ with boundary play a central role in the proofs of all main results of this work, even when specialised to closed manifolds, so we first make good on omitting its definition earlier. \n\nFixing a closed $(d-1)$-manifold $Q$, one replaces $\\smash{\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}}$ with the $\\infty$-groupoid $\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}_Q$ of compact $d$-manifolds with an identification of their boundary with $Q$, and spaces of diffeomorphisms preserving these identifications. The definition \\eqref{equ:presheaf-dfn} of the presheaf $E_M$ still makes sense if $M$ has boundary $Q$ and thus yields a functor $\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}_Q\\rightarrow\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)^\\simeq$, but if $Q\\neq\\varnothing$ then the presheaf $E_M$ carries additional structure. Indeed, stacking cylinders induces an associative algebra structure on the presheaf $E_{Q\\times I}\\in \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ with respect to the symmetric monoidal structure on $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ given by Day convolution, induced by taking disjoint unions in $\\ensuremath{\\icat{D}\\mathrm{isc}}_d$. Similarly, fixing a collar $Q\\times I\\hookrightarrow M$ of the boundary of $M$, the presheaf $E_{M}$ becomes a right-$E_{Q\\times I}$-module. Made precise, this enhances the functor $E\\colon \\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}_Q\\rightarrow\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)^\\simeq$ to a functor \n\\begin{equation}\\label{equ:e-functor-for-left-modules}\n\tE\\colon \\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}_Q\\longrightarrow\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q\\times I}}^\\simeq\n\\end{equation}\nwith target the $\\infty$-groupoid $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q\\times I}}^\\simeq$ of right-$E_{Q\\times I}$-modules. The \\emph{$\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space of a right-$E_{Q\\times I}$-module} $X$ is then defined as the fibre\n\\[\n\tS^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_{Q}(X)\\coloneqq \\mathrm{fib}_{X}\\big(\\ensuremath{\\icat{M}\\mathrm{an}}(d)^{\\cong}_Q\\xra{E}\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q\\times I}}^\\simeq\\big);\n\\]\nthat this recovers the previous definition in the case $Q=\\varnothing$ follows by observing that $E_{\\varnothing\\times I}$ is the monoidal unit. As in the closed case, we abbreviate $\\smash{S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial}(M)\\coloneqq \\smash{S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_{Q}}(E_M)$ if the right-$E_{Q \\times I}$-module $X=E_M$ is induced by a manifold $M$ with identified boundary $\\partial M\\cong Q$. This is the generalisation of $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ for manifolds with boundary in terms of which we stated Theorems~\\ref{bigthm:2-type-invariance}--\\ref{bigthm:nontrivial} above.\n\\subsubsection{Extension to the bordism category}\\label{sec:intr-bordism}\nFor the proofs of these results, we need to generalise the functor \\eqref{equ:e-functor-for-left-modules} further. Given another closed $(d-1)$-manifold $P$, we write $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P,Q}$ for the $\\infty$-groupoid of compact bordisms $W\\colon P\\leadsto Q$ and spaces of diffeomorphisms preserving the identifications of the ends. For such a bordism, the associated presheaf $E_W$ becomes a $(E_{P\\times I},E_{Q\\times I})$-bimodule in $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ and we have a functor \n\\begin{equation}\\label{equ:functor-bimodule}\n\tE\\colon \\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P,Q}\\longrightarrow\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P\\times I},E_{Q\\times I}}^\\simeq\n\\end{equation}\nto the $\\infty$-groupoid $\\smash{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^\\simeq_{E_{P\\times I},E_{Q\\times I}}}$ of $(E_{P\\times I},E_{Q\\times I})$-bimodules, generalising the case $P=\\varnothing$ discussed in the previous subsection. Given another closed $(d-1)$-manifold $R$, one can show that there is a commutative square of $\\infty$-groupoids\n\\[\\begin{tikzcd}[column sep=2cm]\n\t\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P,Q}\\times\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{Q,R}\\rar{(-)\\cup_Q (-)}\\dar[swap]{E\\times E}&\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P,R}\\dar{E}\\\\\n\t\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{P,Q}\\times \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{Q,R}\\rar{(-)\\otimes_{E_{Q\\times I}}(-)}& \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{P,R},\n\\end{tikzcd}\\]\nwhose horizontal functors are induced by gluing bordisms and tensoring bimodules respectively; this is essentially an instance of what is known as $\\otimes$-excision in the theory of factorisation homology. These squares suggest that the functors \\eqref{equ:functor-bimodule} might in fact arise as the maps induced on mapping spaces by a functor of $\\infty$-categories\n\\begin{equation}\\label{equ:e-on-compact-bordisms-intro}\n\tE\\colon \\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}\\longrightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1)}\n\\end{equation}\nfrom the $d$-dimensional bordism category to a Morita category whose objects are associative algebras in $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ and whose morphisms are bimodules. This turns out to be the case, but to prove our results, we need even more functoriality. For this, one notes that the presheaf $E_M$ of a manifold makes equal sense if $M$ is noncompact, so \\eqref{equ:e-on-compact-bordisms-intro} ought to extend to a functor \n\\begin{equation}\\label{equ:e-on-noncompact-bordisms-intro}\n\tE\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{(\\infty,2)}\\longrightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,2)}\n\\end{equation}\nof $(\\infty,2)$-categories from a larger bordism category of possibly noncompact manifolds that has codimension $0$ embeddings as $2$-morphisms, not just diffeomorphisms, to a larger Morita category $\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,2)}$ that has morphisms of bimodules as $2$-morphisms, not just invertible ones.\n\nIn \\cref{sec:the-functor}, relying on work of Haugseng \\cite{HaugsengMorita}, we carefully construct such a functor \\eqref{equ:e-on-noncompact-bordisms-intro} of $(\\infty,2)$-categories and show that it can be enhanced to a functor of \\emph{symmetric monoidal} $(\\infty,2)$-categories. As part of \\cref{sec:functor-e-disc-structure}, we show that for (possibly noncompact) bordisms $W,W'\\colon P\\leadsto Q$ one can identify the map between mapping spaces of $2$-morphisms induced by \\eqref{equ:e-on-noncompact-bordisms-intro}\n\\[\n\\begin{tikzcd}[row sep=0.2cm,ar symbol\/.style = {draw=none,\"\\textstyle#1\" description,sloped},\n\tequivalent\/.style = {ar symbol={\\simeq}}]\n\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}}(W,W')\\rar{E}\\arrow[d,equivalent] &\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{P,Q}}(E_W,E_{W'})\\arrow[d,equivalent] \\\\\n\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')\\rar& T_\\infty\\ensuremath{\\mathrm{Emb}}_\\partial(E_W,E_{W'})\n\\end{tikzcd}\n\\]\nwith Goodwillie--Weiss' embedding calculus approximation, so one might view the functor \\eqref{equ:e-on-noncompact-bordisms-intro} as an enhancement of embedding calculus to the level of bordism categories. In particular,\n\n\\begin{enumerate}[label={(\\Roman*)},leftmargin=0.8cm]\n\t\\item \\label{enum:bord-emb-calc} the functor \\eqref{equ:e-on-noncompact-bordisms-intro} of symmetric monoidal $(\\infty,2)$-categories equips embedding calculus with homotopy coherent gluing and disjoint union maps.\n\\end{enumerate}\nThe functor \\eqref{equ:e-on-noncompact-bordisms-intro} and its relation to embedding calculus forms the technical backbone of the proofs of Theorems~\\ref{bigthm:2-type-invariance}--\\ref{bigthm:nontrivial} in the later chapters, whose proof strategies we summarise now.\n\n\\subsubsection{\\cref{bigthm:2-type-invariance}: tangential $2$-type invariance}\\label{sec:intr-2-type-invariance}\nThe functor \\eqref{equ:e-on-compact-bordisms-intro} in particular extends the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space of a manifold $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ to a space-valued functor of $\\infty$-categories \n\\begin{equation}\\label{equ:functor-on-nullbordism-cat-intro}\n\tS_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(-)\\colon \\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}_{\\varnothing\/}\\longrightarrow \\ensuremath{\\catsingle{S}}\n\\end{equation}\ndefined on the $\\infty$-category of null bordisms, i.e.\\,the undercategory of $\\varnothing\\in \\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}$. Relying on the relation to embedding calculus via \\eqref{equ:e-on-noncompact-bordisms-intro}, a version of an isotopy extension theorem for embedding calculus due to Knudsen--Kupers \\cite{KnudsenKupers}, and Goodwillie--Klein's above mentioned convergence theorem, we show that the functor \\eqref{equ:functor-on-nullbordism-cat-intro} sends a bordism $W\\colon P\\leadsto Q$ to an equivalence if $W$ can be built from a collar on $P$ by attaching handles of index $\\ge3$. This leads to a proof of \\cref{bigthm:2-type-invariance}, since it turns out that the value of \\emph{any} functor of the form \\eqref{equ:functor-on-nullbordism-cat-intro} with this property depends up to equivalence only on the tangential $2$-type. This is an instance of \n\\begin{enumerate}[label={(\\Roman*)},leftmargin=0.8cm,resume]\n\t\\item \\label{enum:general-k-invariance} a general tangential $k$-type invariance result for the values of certain functors on the category $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}_{\\varnothing\/}$ of null bordisms.\n\t\\end{enumerate}\nThe proof of \\ref{enum:general-k-invariance} amounts to a sequence of surgery arguments that we became aware of through the literature on the space of metrics of positive scalar curvature, in particular \\cite{EbertRWbordism,EbertWiemeler}.\n\n\\subsubsection{\\cref{bigthm:infinite-loop-space}: infinite loop space}\\label{sec:intr-infinite-loop-space}\nTo construct an infinite loop space structure on $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$, we first use the tangential $2$-type invariance to show that it suffices to consider manifolds of the form $M=P\\times D^{2n}$ for $P$ a closed manifold and $2n\\ge4$. From the definition\n\\begin{equation}\\label{equ:map-defining-disc-intro}\n\tS^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(P\\times D^{2n})=\\mathrm{fib}_{E_{P\\times D^{2n}}}\\big(\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P\\times S^{2n-1}}\\xra{E}\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P\\times S^{2n-1}\\times I}}^\\simeq\\big),\n\\end{equation}\nit is clear that it suffices to prove that the right-hand map is a map of infinite loop spaces. After restriction to certain path-components that does not affect the fibre, this is what we do. More precisely, in the target, we restrict to modules equivalent to $\\smash{E_{P\\times W_{g,1}}}$ for $g\\ge0$ where $W_{g,1}$ is short for the bordism $(S^n\\times S^n)^{\\sharp g}\\backslash\\mathrm{int}(D^{2n})\\colon \\varnothing\\leadsto S^{2n-1}$. In the source, we restrict to bordisms whose induced presheaf is equivalent to $E_{P\\times W_{g,1}}$ for $g\\ge0$ as a bimodule. We then use the full coherence provided by the functor \\eqref{equ:e-on-compact-bordisms-intro} to enhance the restricted map to one of algebras over a certain higher-dimensional version $\\ensuremath{\\catsingle{W}}$ of Tillmann's surface operad  \\cite{Tillmann}, constructed out of bordisms of the form $\\sqcup^{k} S^{2n-1}\\leadsto \\sqcup^{l} S^{2n-1}$ for $k,l\\ge0$ that are obtained from the manifolds $W_{g,1}$ by creating more boundary spheres. A variant of this operad has already appeared in work of Basterra--Bobkova--Ponto--Tillmann--Yaekel \\cite{BBPTY} on \\emph{operads with homological stability}. They proved that algebras over this operad are $E_1$-spaces (via a  ``pair-of-pants'' product) which group-complete to infinite loop spaces, the main ingredient being a stable homological stability result of Galatius--Randal-Williams \\cite{GRWII}. Translated to our setting, this implies that the fibre of the group completion of the restricted map is an infinite loop space. Using tangential $2$-type invariance once more, we then show that in this case group completion commutes with taking fibres. This only shows that $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(P\\times D^{2n})$ is an infinite loop space \\emph{after group-completion}, but we also show that this $E_1$-space is already group-complete, using the $s$-cobordism theorem.\n\n\\subsubsection{\\cref{bigthm:nontrivial}: nontriviality}\\label{sec:intr-nontriviality}\nTo show that $\\smash{S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial}(M)$ is nontrivial for all compact spin manifolds $M$ of dimension $d\\ge5$, we first reduce to the case $M=D^d$ using tangential $2$-type invariance. Boavida de Brito--Weiss' work \\cite{BdBWConf} in the form of the equivalence \\eqref{equ:pedro-michael-equivalence} further reduces this to showing that the fibre $\\mathrm{Aut}(E_d)\/\\mathrm{Top}(d)$ of \\eqref{equ:topd-to-ed-intro} has a nontrivial homotopy group in sufficiently high degree, which we do by showing that the individual homotopy groups of $\\mathrm{Aut}(E_d)$ and $\\mathrm{Top}(d)$ are sufficiently different. While quite a bit is known on the homotopy groups of $\\mathrm{Top}(d)$, especially rationally, so far almost nothing is known about the homotopy groups of $\\mathrm{Aut}(E_d)$ besides for small values of $d$. This is in stark contrast to the automorphism group $\\mathrm{Aut}((E_d)_{\\ensuremath{\\mathbf{Q}}})$ of the \\emph{rationalised} $E_d$-operad, whose homotopy groups have a complete description in terms of graph complexes \u00e0 la Kontsevich due to work of Fresse--Turchin--Willwacher \\cite{FTW}. Thus, to learn something about the homotopy groups of $\\mathrm{Aut}(E_d)$, one could try to study the comparison map $\\mathrm{Aut}(E_d)\\rightarrow \\mathrm{Aut}((E_d)_{\\ensuremath{\\mathbf{Q}}})$ on homotopy groups. This is what we do. More generally, \n\\begin{enumerate}[label={(\\Roman*)},leftmargin=0.8cm,resume]\n\t\\item \\label{enum:rationalisation-operads-intro} we study the effect on homotopy groups of the map $\\mathrm{Map}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})\\rightarrow \\mathrm{Map}(\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}})$ for operads $\\ensuremath{\\catsingle{O}}$ and $\\ensuremath{\\catsingle{P}}$, induced by rationalisation. \n\\end{enumerate}\nFor this, we first use work of G\\\"oppl \\cite{Goppl} to decompose the mapping spaces as a limit of a tower of mapping spaces between truncated operads and show that under mild assumptions, the maps analogous to that in \\ref{enum:rationalisation-operads-intro} between the stages of this tower are componentwise rationalisations. Rationalisation does \\emph{not} commute with sequential limits in general, so this does \\emph{not} imply that the map in \\ref{enum:rationalisation-operads-intro} has the same property. However, we then show that this can only fail in an extreme way, namely when some of the homotopy groups of $\\mathrm{Map}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})$ are uncountable. We also explain similar results for more general localisations and for more general towers of spaces.\n\nApplied to $\\ensuremath{\\catsingle{O}}=\\ensuremath{\\catsingle{P}}=E_d$, this shows that the homotopy groups of $\\mathrm{Aut}(E_d)$ either agree rationally with those of $\\mathrm{Aut}((E_d)_{\\ensuremath{\\mathbf{Q}}})$, as described in Fresse--Turchin--Willwacher's work, or some of them are uncountable. In either case, we can conclude that they are different from that of $\\mathrm{Top}(d)$: in the former by comparing them with known partial computations of the rational homotopy groups of $\\mathrm{Top}(d)$, and in the latter by using that $\\mathrm{Top}(d)$ has countable homotopy groups.\n\n\\subsection*{Acknowledgements} Our thanks go to Fabian Hebestreit and Markus Land for answering several questions on $\\infty$-categories, to Rune Haugseng and Claudia Scheimbauer for helpful conversations on their work, to Calista Bernard for sharing her view on manifold calculus and bordism categories, and to Oscar Randal-Williams for general discussions.\n\nMK was partially funded by the ERC under the European Union's Horizon 2020 research and innovation programme (grant agreement No.\\,756444), and partially by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044 \u2013390685587, Mathematics M\u00fcnster: Dynamics\u2013Geometry\u2013Structure. \n\nAK acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) [funding reference number 512156 and 512250], as well as the Research Competitiveness Fund of the University of Toronto at Scarborough.\n\nThis material is partially based on work supported by the Swedish Research Council under grant no.\\,2016-06596 while the authors were in residence at Institut Mittag-Leffler in Djursholm, Sweden during the semester \\emph{Higher algebraic structures in algebra, topology and geometry}. \n\n\n\\section{$\\infty$-categorical preliminaries} \\label{sec:preliminaries}\nExcept for the final two sections (see \\cref{conv:no-more-infty}), we work in the setting of $\\infty$-categories. This section---which may be skipped on first reading and referred back to when necessary---serves to establish some notation and to recall definitions and facts used in later sections, as well as to prove a few technical results that we could not find in the literature. The topics are:\n\n\\begin{minipage}[c]{\\textwidth}\n\\begin{multicols}{2}\n\\begin{enumerate}[leftmargin=0.1cm]\n\t\\item[\\ref{sec:conventions}] Conventions.\n\t\\item[\\ref{sec:scat-vs-qcat}] The coherent nerve.\n\t\\item[\\ref{sec:straightening}] Cocartesian fibrations.\n\t\\item[\\ref{section:delta-cut}] The categories $\\Delta$, $\\ensuremath{\\cat{Cut}}$, and $\\ensuremath{\\cat{Fin}}_*$.\n\t\\item[\\ref{sec:cat-objects}] Category and monoid objects.\n\t\\item[\\ref{sec:presheaf-category}] Presheaves and the Yoneda embedding.\n\t\\item[\\ref{sec:gen-infty-operads}] $\\infty$-operads and generalised $\\infty$-operads.\n\t\\item[\\ref{sec:assalg-bimodules}] Associative algebras and bimodules.\n\t\\item[\\ref{sec:haugseng-morita}] Haugseng's Morita category.\n\t\\item[\\ref{sec:span-cospan-cats}] Span and cospan categories.\n\\end{enumerate}\n\\end{multicols}\n\\end{minipage}\n\n\\subsection{Conventions}\\label{sec:conventions}\nUnless mentioned otherwise we follow the conventions and notation of Lurie \\cite{LurieHTT,LurieHA}. In particular:\n\\begin{itemize}\n\t\\item An \\emph{$\\infty$-category} is a \\emph{quasi-category} \\cite[1.1.2.4]{LurieHTT}. The \\emph{$\\infty$-category of $\\infty$-categories} $\\gls*{catinf}$ is the coherent nerve $\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}\\coloneqq N_{\\mathrm{coh}}(\\ensuremath{\\cat{Cat}_\\infty})$ of the $\\ensuremath{\\cat{Kan}}$-enriched category $\\ensuremath{\\cat{Cat}_\\infty}$ of small $\\infty$-categories \\cite[3.0.0.1]{LurieHTT}. We consider 1-categories as $\\infty$-categories via their nerve.\n\t\\item A \\emph{space} is a Kan complex. If topological spaces appear, we implicitly replace them by their singular simplicial sets. The category of simplicial sets is denoted $\\ensuremath{\\cat{S}}$ and the full subcategory of Kan-complexes by $\\ensuremath{\\cat{Kan}} \\subset \\cat{S}$. Both are enriched over themselves. The \\emph{$\\infty$-category of spaces} $\\gls*{sinf}$ is the coherent nerve $\\ensuremath{\\catsingle{S}} \\coloneqq N_{\\mathrm{coh}}(\\ensuremath{\\cat{Kan}})$  \\cite[1.2.16.1]{LurieHTT}.\n\\end{itemize}\n\n\\noindent We use the following notational conventions:\n\\begin{itemize}\n\t\\item The letters $\\ensuremath{\\catsingle{A}}$, $\\ensuremath{\\catsingle{B} }$, $\\ensuremath{\\catsingle{C}}$, $\\ldots$ typically stand for $\\infty$-categories, whereas the letters $\\ensuremath{\\cat{A}}$, $\\ensuremath{\\cat{B}}$, $\\ensuremath{\\cat{C}}$, $\\ldots$ usually stand for $\\ensuremath{\\cat{S}}$-enriched, $\\ensuremath{\\cat{Kan}}$-enriched, or 1-categories.\n\t\\item Given an $\\infty$-category $\\ensuremath{\\catsingle{C}}$ and object $c$ of $\\ensuremath{\\catsingle{C}}$, $\\ensuremath{\\catsingle{C}}_{c\/}^\\mathrm{op}$ is short for $(\\ensuremath{\\catsingle{C}}_{c\/})^\\mathrm{op}$ and similarly $\\ensuremath{\\catsingle{C}}^\\mathrm{op}_{\/c}$ is short for $(\\ensuremath{\\catsingle{C}}_{\/c})^\\mathrm{op}$. In other words, slices are taken \\emph{before} opposite categories.\n\\end{itemize}\n\n\\subsection{The coherent nerve and the homotopy category}\\label{sec:scat-vs-qcat} \n\\label{sec:coherent-nerve-props}\nThe \\emph{coherent nerve}  $\\gls*{ncoh} \\colon \\ensuremath{\\cat{sCat}}\\rightarrow \\cat{S}$ is an $\\ensuremath{\\cat{S}}$-enriched functor from the $1$-category $\\ensuremath{\\cat{sCat}}$ of $\\ensuremath{\\cat{S}}$-enriched categories to the $1$-category of simplicial sets \\cite[1.1.5]{LurieHTT}. Some of its properties are:\n\\begin{enumerate}\n\t\\item \\label{enum:bergner-model-structure} It is the right-adjoint in a Quillen equivalence \\cite[2.2.5.1]{LurieHTT}, where $\\ensuremath{\\cat{sCat}}$ is equipped with the Bergner model structure whose \n\t\\begin{enumerate}\n\t\t\\item fibrant objects are $\\ensuremath{\\cat{Kan}}$-enriched categories \\cite[A.3.2.24]{LurieHTT},\n\t\t\\item weak equivalences are \\emph{Dwyer--Kan equivalences}, so simplicial functors that induce weak homotopy equivalences on each mapping space and are an equivalence (of $1$-categories) on homotopy categories \\cite[A.3.2.4]{LurieHTT},\n\t\t\\item fibrations are simplicial functors that are Kan fibrations on each mapping space and isofibrations on homotopy categories \\cite[A.3.2.24, A.3.2.25]{LurieHTT},\n\t\\end{enumerate}\n\tand $\\cat{S}$ is equipped with the Joyal model structure of which we only need to know that its fibrant objects are precisely $\\infty$-categories \\cite[2.4.6.1]{LurieHTT}. In particular, the coherent nerve of a $\\ensuremath{\\cat{Kan}}$-enriched category is an $\\infty$-category.\n\t\\item \\label{enum:objects-morphisms}Taking coherent nerves preserves objects and morphisms, in the sense that the $0$- and $1$-simplices of $N_{\\mathrm{coh}}(\\cat{C})$ are the sets of objects and morphisms of $\\cat{C}$ \\cite[p.\\,23]{LurieHTT}.\n\t\\item \\label{enum:simplicial-mapping-spaces} Taking coherent nerves preserves mapping spaces of $\\ensuremath{\\cat{Kan}}$-enriched categories in that for a $\\ensuremath{\\cat{Kan}}$-enriched category $\\cat{C}$ we have $\\mathrm{Map}_{\\cat{C}}(c,c')\\simeq \\mathrm{Map}_{N_{\\mathrm{coh}}(\\cat{C})}(c,c')$ \\cite[2.2]{LurieHTT}.\n\t\\item \\label{enum:coherent-opposite} There is a natural equivalence $N_\\mathrm{coh}(\\cat{C}^\\mathrm{op}) \\simeq N_\\mathrm{coh}(\\cat{C})^\\mathrm{op}$. This is a consequence of the natural isomorphisms  $\\mathfrak{C}([n]^\\mathrm{op}) \\cong \\mathfrak{C}([n])^\\mathrm{op}$, where $\\mathfrak{C}(-)$ is the left adjoint to $N_{\\mathrm{coh}}(-)$.\n\t\\item \\label{enum:coherent-functor} There is a canonical map $N_\\mathrm{coh}(\\ensuremath{\\mathrm{Fun}}(\\cat{C},\\cat{D})) \\to \\ensuremath{\\mathrm{Fun}}(N_\\mathrm{coh}(\\cat{C}),N_\\mathrm{coh}(\\cat{D}))$ obtained by appling $N_\\mathrm{coh}$ to the evaluation $\\ensuremath{\\mathrm{Fun}}(\\cat{C},\\cat{D}) \\times \\cat{C} \\rightarrow \\cat{D}$, using that as a right adjoint $N_\\mathrm{coh}(-)$ preserves products to get $N_\\mathrm{coh}(\\ensuremath{\\mathrm{Fun}}(\\cat{C},\\cat{D})) \\times N_\\mathrm{coh}(\\cat{C}) \\to N_\\mathrm{coh}(\\cat{D})$, and adjoining over $N_\\mathrm{coh}(\\cat{C})$.\\end{enumerate}\nRestricting $N_\\mathrm{coh}$ to $\\cat{Cat} \\subset \\ensuremath{\\cat{sCat}}$ gives a fully faithful functor of $1$-categories from ordinary $1$-categories to $\\infty$-categories. Applying $N_{\\mathrm{coh}}$, we obtain a functor $\\ensuremath{\\mathrm{Cat}}\\rightarrow \\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$ of $\\infty$-categories. This has a left-adjoint $\\gls*{homcat} \\colon \\ensuremath{\\icat{C}\\mathrm{at}_\\infty} \\to \\ensuremath{\\mathrm{Cat}}$ that assigns an $\\infty$-category its \\emph{homotopy category}. As described in \\cite[1.2.3]{LurieHTT}, $h\\ensuremath{\\catsingle{C}}$ has the same objects as $\\ensuremath{\\catsingle{C}}$, morphism sets given by $\\pi_0$ of the mapping spaces in $\\ensuremath{\\catsingle{C}}$, and composition is induced by the composition maps of mapping spaces. Some of its further properties are:\n\\begin{enumerate}\n\t\\item The functor $h$ preserves products.\n\t\\item The functor $h$ preserves pullbacks if one of the maps is between 1-categories.\n\t\\item The functor $h$ preserves cocartesian morphisms when the target is an 1-category.\n\\end{enumerate}\nThese follow from the facts that taking mapping spaces in $\\infty$-categories preserves pullbacks, and that taking components preserves pullbacks in $\\ensuremath{\\catsingle{S}}$ whose bottom right corner is discrete. \n\n\\subsection{Cocartesian fibrations} \\label{sec:straightening}\nLurie's \\emph{straightening equivalence} \\cite[3.2]{LurieHTT}\n\\begin{equation}\\label{equ:st-un}\n\t\t\\ensuremath{\\mathrm{Fun}}(\\icat{C},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}) \\simeq \\ensuremath{\\mathrm{Cocart}}(\\icat{C})\n\\end{equation}\nidentifies the $\\infty$-category $\\ensuremath{\\mathrm{Fun}}(\\icat{C},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$ for an $\\infty$-category $\\ensuremath{\\catsingle{C}}$ with the  $\\infty$-category of \\emph{cocartesian fibrations}, which is the sub $\\infty$-category $\\ensuremath{\\mathrm{Cocart}}(\\ensuremath{\\catsingle{C}})\\subset (\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})_{\/\\ensuremath{\\catsingle{C}}}$ with objects \\emph{cocartesian fibrations} with target $\\ensuremath{\\catsingle{C}}$ and whose morphisms  \\emph{maps of cocartesian fibrations}, in the following sense:\n\\begin{dfn}\\label{dfn:cocartesian-fibration}Let $\\varphi\\colon \\ensuremath{\\catsingle{E}} \\rightarrow \\ensuremath{\\catsingle{B} } $ be a functor between $\\infty$-categories. \n\t\\begin{enumerate}\n\t\t\\item \n\t\tA morphism $f\\colon e\\rightarrow e'$ in $\\ensuremath{\\catsingle{E}} $ is \\emph{$\\varphi$-cocartesian} if for every $x\\in \\ensuremath{\\catsingle{E}} $ the square\n\t\t\\[\n\t\t\\begin{tikzcd}\n\t\t\t\\mathrm{Map}_{\\icat{E}}(e',x)\\rar{f^*}\\dar{\\varphi}&\\mathrm{Map}_{\\icat{E}}(e,x)\\dar{\\varphi}\\\\\n\t\t\t\\mathrm{Map}_{\\icat{B}}(\\varphi(e'),\\varphi(x))\\rar{\\varphi(f)^*}&\\mathrm{Map}_{\\icat{B}}(\\varphi(e),\\varphi(x))\n\t\t\\end{tikzcd}\n\t\t\\]\n\t\tis homotopy cartesian.\n\t\t\\item The functor $\\varphi$ is a \\emph{cocartesian fibration} if for every object $e\\in\\ensuremath{\\catsingle{E}} $ and morphism $f\\colon \\varphi(e)\\rightarrow b$ there exists a \\emph{cocartesian lift} of $f$, i.e.\\,a $\\varphi$-cocartesian morphism $\\tilde{f}\\colon e\\rightarrow \\tilde{b}$ for some $\\tilde{b}$ in $\\ensuremath{\\catsingle{E}} $ such that $\\varphi(\\tilde{f})=f$.\n\t\t\\item A \\emph{map of cocartesian fibrations} from $\\varphi\\colon \\ensuremath{\\catsingle{E}} \\rightarrow \\ensuremath{\\catsingle{B} } $ to $\\varphi'\\colon \\ensuremath{\\catsingle{E}} '\\rightarrow \\ensuremath{\\catsingle{B} } $ is a functor $\\ensuremath{\\catsingle{E}} \\rightarrow \\ensuremath{\\catsingle{E}} '$ over $\\ensuremath{\\catsingle{B} } $ that sends $\\varphi$-cocartesian morphisms to $\\varphi'$-cocartesian morphisms.\n\t\\end{enumerate}\n\\end{dfn}\nGiven a cocartesian fibration $\\varphi\\colon \\ensuremath{\\catsingle{E}} \\rightarrow \\ensuremath{\\catsingle{B} } $ and an object $b\\in\\ensuremath{\\catsingle{B} } $, one writes $\\ensuremath{\\catsingle{E}} _b\\in\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$ for the fibre of $\\varphi$ at $b$. Under the straightening equivalence \\eqref{equ:st-un}, this corresponds to the value at $b$ of the associated functor $\\ensuremath{\\catsingle{B} } \\rightarrow \\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$. Moreover, the value of this functor on a morphism $b\\rightarrow b'$ in $\\ensuremath{\\catsingle{B} } $ corresponds to a functor $\\ensuremath{\\catsingle{E}}_b\\rightarrow \\ensuremath{\\catsingle{E}}_b'$ induced by choosing cocartesian lifts of $b\\rightarrow b'$. \n\n\\begin{rem}\\label{fact:cocart-from-s-cat}\n\t \\cref{dfn:cocartesian-fibration} makes equal sense for a functor $\\varphi\\colon \\cat{E}\\rightarrow \\cat{B}$ of $\\ensuremath{\\cat{Kan}}$-enriched categories. In view of the natural equivalence  $\\mathrm{Map}_{\\cat{C}}(c,c')\\simeq \\mathrm{Map}_{N_{\\mathrm{coh}}(\\cat{C})}(c,c')$, one sees that a morphism $f\\colon e\\rightarrow e'$ in $\\cat{E}$ is $\\varphi$-cocartesian if and only if is $N_{\\mathrm{coh}}(\\varphi)$-cocartesian.\n\\end{rem}\n\\glsadd{cocartpush}\nGiven a cocartesian fibration $\\varphi\\colon \\ensuremath{\\catsingle{E}} \\rightarrow \\ensuremath{\\catsingle{B} } $ and an $\\infty$-category $\\ensuremath{\\catsingle{C}}$, the functor $\\varphi_*\\colon \\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\catsingle{C}},\\ensuremath{\\catsingle{E}}) \\rightarrow \\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\catsingle{C}},\\ensuremath{\\catsingle{B} })$ is again a cocartesian fibration  \\cite[3.1.2.1]{LurieHTT}. In particular, given a functor $f \\colon \\ensuremath{\\catsingle{C}}\\rightarrow \\ensuremath{\\catsingle{E}}$ and a morphism $\\eta \\colon (\\varphi\\circ f)\\rightarrow *_b$ to the constant functor $*_b\\colon \\ensuremath{\\catsingle{C}}\\rightarrow \\ensuremath{\\catsingle{B} }$ with value $b\\in B$ (equivalently, an extension of $(\\varphi\\circ f)$ to a functor $\\ensuremath{\\catsingle{C}}^\\rhd \\rightarrow \\ensuremath{\\catsingle{B} }$ on the right-cone whose value at the cone point is $b$), we can use that $\\varphi_*$ is a cocartesian fibration to obtain a cocartesian lift to a functor $f_!\\colon \\ensuremath{\\catsingle{C}}\\rightarrow \\ensuremath{\\catsingle{B} }_b$ into  the fibre over $b$. The functor $f_!$ is called a \\emph{cocartesian pushforward} of $f$ along $\\eta$.\n\n\n\\subsection{The categories $\\ensuremath{\\cat{Fin}}_*$, $\\Delta$, and $\\ensuremath{\\cat{Cut}}$} \\label{section:delta-cut}\nRecall the 1-category \\gls*{fin} of pointed finite sets and pointed maps in between, with skeleton given by $\\gls*{langlerangle} = \\{1,\\ldots,p,\\ast\\}$ for $p \\geq 0$. We write $\\langle \\mathring{p} \\rangle \\coloneqq \\langle p \\rangle \\setminus \\{\\ast\\}$ for the \\emph{interior} of $\\langle p \\rangle$. Three special types of morphisms are relevant for us: $\\alpha\\colon \\langle p \\rangle \\rightarrow \\langle q \\rangle$ is \n\\begin{enumerate}\n\t\\item \\emph{active} if it satisfies $\\alpha^{-1}(\\ast)=\\{\\ast\\}$, \n\t\\item \\emph{inert} if $\\alpha^{-1}(i)$ consists of a single element for all $i \\in \\langle \\mathring{q} \\rangle$,\n\t\\item \\emph{Segal} if it agrees with $\\rho_i\\colon \\langle p \\rangle \\rightarrow \\langle 1 \\rangle$ for some $1\\le i\\le p$ where $\\rho_i(i)=1$ and $\\rho(j)=\\ast$ otherwise. Note this is equivalent to being inert with target $\\langle 1 \\rangle$.\n\\end{enumerate}\nA closely related 1-category is the \\emph{simplex category}  $\\gls*{delta}$ of non-empty finite totally ordered sets and weakly order-preserving maps between them. We mostly work with its skeleton given by $\\gls*{brpbr}=(0<1<\\ldots<p)$ for $p\\ge0$. The wide subcategory of injective maps is denoted $\\gls*{deltainj} \\subset \\Delta$. Four special types of morphisms are relevant for us: a morphism $\\alpha\\colon [p]\\rightarrow [q]$ is \n\\begin{enumerate}\n\t\\item \\emph{active} if it satisfies $\\alpha(0)=0$ and $\\alpha(p)=q$, \n\t\\item \\emph{cellular} if $\\alpha(i+1)\\le \\alpha(i)+1$ for all $i$,\n\t\\item \\emph{inert} if it is the inclusion of a subinterval, i.e.\\,$\\alpha(i)=\\alpha(0)+i$ for all $i$, and\n\t\\item \\emph{Segal} if it agrees with $\\rho_i\\colon [1]\\rightarrow [q]$ for some $1\\le i\\le q$ where $\\rho_i(0)=i-1$ and $\\rho(1)=i$. Note this is equivalent to being inert with domain $[1]$.\n\\end{enumerate}\nOccasionally we work with a different model for $\\Delta^{\\mathrm{op}}$, given as follows. For $p\\ge0$ we write $\\gls*{lbrrbr}$ for the totally ordered set $(L<1<\\ldots< p<R)$ and call $L$ and $R$ the \\emph{left end} and \\emph{right end} of $\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis$ respectively. The sets $\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis$ for $p\\ge0$ form the objects of the category $\\gls*{cut}$ whose morphisms are weakly order-preserving maps that are the identity on ends. There is an isomorphism \\begin{equation}\\label{equ:cut-iso}\n\tc\\colon \\Delta^\\mathrm{op}\\xlra{\\cong} \\ensuremath{\\cat{Cut}}\n\\end{equation} \nthat sends $[p]$ to $\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis$ and a morphism $\\alpha\\colon [p]\\rightarrow [q]$ to the morphism $c(\\alpha)\\colon \\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis\\rightarrow  \\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis$ given by\n\\begin{align*}\n\t{i}&\\longmapsto \\begin{cases}L& i\\le  \\alpha(0),\\\\\n\t\tj& \\exists j\\colon i\\in [\\alpha(j-1)+1,\\alpha(j)],\\\\\n\t\tR& i>\\alpha(p).\\end{cases}\n\\end{align*}\nThis isomorphism maps $\\smash{\\Delta_\\mathrm{inj}^\\mathrm{op}}\\subset \\Delta^\\mathrm{op}$ isomorphically onto the wide subcategory $\\gls*{cutsur} \\subset \\ensuremath{\\cat{Cut}}$ of surjective maps. Introducing the notation $\\llparenthesis\\hspace{.1em} \\mathring{p}\\hspace{.1em}\\rrparenthesis\\coloneqq \\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\\backslash\\{L,R\\}$ for the \\emph{interior} of $\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis$, a morphism $\\alpha\\colon \\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis \\rightarrow \\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis$, when considered as a morphism $c^{-1}(\\alpha)\\colon [p]\\rightarrow [q]$ in $\\Delta$, is \n\\begin{enumerate}\n\t\\item \\emph{active} if $\\alpha^{-1}\\llparenthesis\\hspace{.1em} \\mathring{p}\\hspace{.1em}\\rrparenthesis=\\llparenthesis\\hspace{.1em} \\mathring{q}\\hspace{.1em}\\rrparenthesis$ (we omit the parentheses in $\\alpha^{-1}(\\llparenthesis\\hspace{.1em} \\mathring{p}\\hspace{.1em}\\rrparenthesis)$ for legibility),\n\t\\item \\emph{cellular} if the restriction $\\alpha\\colon \\alpha^{-1}\\llparenthesis\\hspace{.1em} \\mathring{p}\\hspace{.1em}\\rrparenthesis\\rightarrow \\llparenthesis\\hspace{.1em} \\mathring{p}\\hspace{.1em}\\rrparenthesis$ is injective,\n\t\\item \\emph{inert} if the restriction $\\alpha\\colon \\alpha^{-1}\\llparenthesis\\hspace{.1em} \\mathring{p}\\hspace{.1em}\\rrparenthesis\\rightarrow \\llparenthesis\\hspace{.1em} \\mathring{p}\\hspace{.1em}\\rrparenthesis$ is bijective,\n\t\\item \\emph{Segal} if it agrees with $\\rho_i'\\colon\\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis \\rightarrow \\llparenthesis\\hspace{.1em} 1\\hspace{.1em}\\rrparenthesis$ for some $1\\le i\\le q$ where $\\rho_i'(j)=L$ if $j<i$, $\\rho_i'(j)=1$ if $j=i$, and  $\\rho_i'(j)=R$ if $j>i$.\n\\end{enumerate}\n\n\n\\begin{figure}\n\t\\begin{tikzpicture}[scale=.6]\n\t\t\n\t\t\\begin{scope}\n\t\t\t\\node at (-2,0) {$[5]$};\n\t\t\t\\node at (-2,-2) {$[6]$};\n\t\t\t\\draw[->,shorten >=.3cm,shorten <=.3cm] (-2,0) -- (-2,-2);\n\t\t\t\n\t\t\t\\foreach \\i in {0,...,5}\n\t\t\t{\n\t\t\t\t\\node [Mahogany] at (\\i,0) {$\\bullet$};\n\t\t\t\t\\node at (\\i,.5) {\\tiny $\\i$};\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,6}\n\t\t\t{\n\t\t\t\t\\node [Mahogany] at ({\\i-.5},-2) {$\\bullet$};\n\t\t\t\t\\node at ({\\i-.5},-2.5) {\\tiny $\\i$};\n\t\t\t}\n\t\t\t\\draw[Mahogany,->,shorten >=.2cm,shorten <=.2cm] (0,0) -- ({1-.5},-2);\n\t\t\t\\draw[Mahogany,->,shorten >=.2cm,shorten <=.2cm] (1,0) -- ({1-.5},-2);\n\t\t\t\\draw[Mahogany,->,shorten >=.2cm,shorten <=.2cm] (2,0) -- ({2-.5},-2);\n\t\t\t\\draw[Mahogany,->,shorten >=.2cm,shorten <=.2cm] (3,0) -- ({4-.5},-2);\n\t\t\t\\draw[Mahogany,->,shorten >=.2cm,shorten <=.2cm] (4,0) -- ({5-.5},-2);\n\t\t\t\\draw[Mahogany,->,shorten >=.2cm,shorten <=.2cm] (5,0) -- ({6-.5},-2);\n\t\t\\end{scope}\n\t\t\n\t\t\\begin{scope}[xshift=11cm]\n\t\t\t\\node at (-2.5,0) {$\\llparenthesis\\hspace{.1em} 5 \\hspace{.1em}\\rrparenthesis$};\n\t\t\t\\node at (-2.5,-2) {$\\llparenthesis\\hspace{.1em} 6 \\hspace{.1em}\\rrparenthesis$};\n\t\t\t\\draw[<-,shorten >=.3cm,shorten <=.3cm] (-2.5,0) -- (-2.5,-2);\n\t\t\t\n\t\t\t\\node [Mahogany] at (-.5,0) {$\\ast$};\n\t\t\t\\node at (-.5,.4) {\\tiny $L$};\n\t\t\t\\node [Mahogany] at (5.5,0) {$\\ast$};\n\t\t\t\\node at (5.5,.4) {\\tiny $R$};\n\t\t\t\\foreach \\i in {1,...,5}\n\t\t\t{\n\t\t\t\t\\draw[Mahogany,very thick,shorten >=.1cm,shorten <=.1cm] ({\\i-1},0) -- (\\i,0);\n\t\t\t\t\\node at ({\\i-.5},.4) {\\tiny $\\i$};\n\t\t\t}\n\t\t\t\n\t\t\t\\node [Mahogany] at (-1,-2) {$\\ast$};\n\t\t\t\\node at (-1,-2.4) {\\tiny $L$};\n\t\t\t\\node [Mahogany] at (6,-2) {$\\ast$};\n\t\t\t\\node at (6,-2.4) {\\tiny $R$};\n\t\t\t\\foreach \\i in {1,...,6}\n\t\t\t{\n\t\t\t\t\\draw[Mahogany,very thick,shorten >=.1cm,shorten <=.1cm] ({\\i-1.5},-2) -- ({\\i-.5},-2);\n\t\t\t\t\\node at ({\\i-1},-2.4) {\\tiny $\\i$};\n\t\t\t}\n\t\t\t\\draw[dotted,shorten >=.2cm,shorten <=.2cm] (0,0) -- ({1-.5},-2);\n\t\t\t\\draw[dotted,shorten >=.2cm,shorten <=.2cm] (1,0) -- ({1-.5},-2);\n\t\t\t\\draw[Mahogany,<-,shorten >=.15cm,shorten <=.15cm] (1.5,0) -- (1,-2);\n\t\t\t\\draw[dotted,shorten >=.2cm,shorten <=.2cm] (2,0) -- ({2-.5},-2);\n\t\t\t\\draw[Mahogany,<-,shorten >=.15cm,shorten <=.15cm] (2.5,0) -- (2,-2);\n\t\t\t\\draw[Mahogany,<-,shorten >=.15cm,shorten <=.15cm] (2.5,0) -- (3,-2);\n\t\t\t\\draw[dotted,shorten >=.2cm,shorten <=.2cm] (3,0) -- ({4-.5},-2);\n\t\t\t\\draw[Mahogany,<-,shorten >=.15cm,shorten <=.15cm] (3.5,0) -- (4,-2);\n\t\t\t\\draw[dotted,shorten >=.2cm,shorten <=.2cm] (4,0) -- ({5-.5},-2);\n\t\t\t\\draw[Mahogany,<-,shorten >=.15cm,shorten <=.15cm] (4.5,0) -- (5,-2);\n\t\t\t\\draw[dotted,shorten >=.2cm,shorten <=.2cm] (5,0) -- ({6-.5},-2);\n\t\t\\end{scope}\n\t\\end{tikzpicture}\n\t\\caption{The isomorphism \\eqref{equ:cut-iso} between $\\Delta^\\mathrm{op}$ (on the left) and $\\ensuremath{\\cat{Cut}}$ (on the right, but we omitted the elements that map to $L$ or $R$). The morphism indicated is not active, cellular, inert, or Segal.}\n\t\\label{figure:delta-to-cut}\n\\end{figure}\n\n\n\\begin{rem}We think of $i \\in \\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis$ as the ``cut'' between $i-1$ and $i$ in $[p]$, and observe that $\\alpha \\colon [p] \\to [q]$ induces a map the other way between these cuts; see \\cref{figure:delta-to-cut} for an example.\n\\end{rem}\n\nThe three 1-categories $\\ensuremath{\\cat{Fin}}_*$, $\\Delta$, and $\\ensuremath{\\cat{Cut}}$ are related by a sequence of functors \n\\begin{equation}\\label{equ:delta-to-fin}\n\t\\Delta^{\\mathrm{op}} \\longrightarrow\\ensuremath{\\cat{Cut}} \\longrightarrow \\ensuremath{\\cat{Fin}}_*\n\\end{equation}\nwhere the first arrow is the isomorphism \\eqref{equ:cut-iso}, and the second arrow is obtained by identifying the left and right ends $L$ and $R$ of objects in $\\ensuremath{\\cat{Cut}}$ and forgetting that morphisms are order-preserving.\n\n\n\n\\subsection{Category and monoid objects}\\label{sec:cat-objects} Fix an $\\infty$-category $\\icat{C}$ with finite limits. \n\n\\subsubsection{Category objects and monoid objects}\nA \\emph{category object} in $\\icat{C}$ is a simplicial object $X\\in \\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}},\\icat{C})$ satisfying the \\emph{Segal condition}, i.e.\\,the map \n\t\t\\begin{equation}\\label{equ:segal-maps}\n\t\t\tX_{[p]} \\longrightarrow X_{[1]} \\times_{X_{[0]}} \\cdots \\times_{X_{[0]}} X_{[1]}\n\t\t\\end{equation}\n\t\tinduced by the Segal maps $\\rho_i\\colon [1] \\to [p]$ for $1\\le i\\le p$, is an equivalence for all $p\\ge0$. We call $X_{[1]}$ the \\emph{underlying object} of $X$. A \\emph{monoid object} is a category object $X$ for which the map $X_{[0]}\\rightarrow *$ to the terminal object is an equivalence; equivalently it is a simplicial object for which the analogues of the maps \\eqref{equ:segal-maps} with pullbacks replaced by products are equivalences. We write \n\\[\n\t\\gls*{caticat} \\subset \\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}},\\icat{C})\\quad\\text{and}\\quad\n\t\\gls*{monicat} \\subset \\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}},\\icat{C})\n\\] \nfor the full subcategories of category objects and monoid objects. Replacing simplicial by semisimplicial objects in this definition yields the categories \n\\[\n\t\\gls*{catnuicat} \\subset \\ensuremath{\\mathrm{Fun}}(\\Delta_\\mathrm{inj}^{\\mathrm{op}},\\icat{C})\\quad\\text{and}\\quad\n\t\\gls*{monnuicat} \\subset \\ensuremath{\\mathrm{Fun}}(\\Delta_\\mathrm{inj}^{\\mathrm{op}},\\icat{C})\n\\]\nof \\emph{non-unital category objects} and \\emph{non-unital monoid objects}. \n\n\\subsubsection{Commutative monoid objects} \nWe may replace the role of the category $\\Delta^{\\mathrm{op}}$ in the definition of a monoid object with $\\ensuremath{\\cat{Fin}}_*$ to arrive at the notion of a \\emph{commutative monoid object}: a functor $X\\in\\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{Fin}}_*,\\ensuremath{\\catsingle{C}})$ for which the maps $X_{\\langle p \\rangle}\\rightarrow X_{\\langle 1 \\rangle}\\times\\ldots\\times X_{\\langle 1 \\rangle}$ induced by the Segal maps $\\rho_i\\colon \\langle p \\rangle\\rightarrow \\langle 1 \\rangle$ for $1\\le i\\le p$ are equivalences for all $p\\ge0$. These span the full subcategory\n\\[\n\t\\gls*{cmonnuicat}\\subset\\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{Fin}}_*,\\ensuremath{\\catsingle{C}})\n\\] \nof commutative monoid objects. Precomposition with the composition $\\Delta^\\mathrm{op} \\to \\ensuremath{\\cat{Fin}}_*$ of \\eqref{equ:delta-to-fin} induces a functor $\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\catsingle{C}})\\rightarrow\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{C}})$ that ``forgets commutativity''.\n\n\\begin{rem}\\label{rem:comm-mon-iterative}\nThere is a different perspective on commutative monoid objects in the form of an equivalence of $\\infty$-categories $\\ensuremath{\\mathrm{Mon}}_{\\infty}(\\ensuremath{\\catsingle{C}})\\simeq \\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\catsingle{C}})$ where $\\ensuremath{\\mathrm{Mon}}_{\\infty}(\\ensuremath{\\catsingle{C}})$ is the limit in $\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$ \\[\\ensuremath{\\mathrm{Mon}}_{\\infty}(\\ensuremath{\\catsingle{C}})\\simeq \\lim\\big(\\cdots \\rightarrow \\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{C}})))\\rightarrow \\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{C}}))\\rightarrow \\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{C}})\\rightarrow \\ensuremath{\\catsingle{C}}\\big)\\]  over the maps induced $\\mathrm{ev}_{[1]}\\colon \\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{C}})\\rightarrow \\ensuremath{\\catsingle{C}}$ (combine \\cite[Proposition 10.11]{HaugsengSpans} with \\cite[5.1.1.5, 2.4.2.5]{LurieHA}). In particular, there is an equivalence $\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{C}}))\\simeq \\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\catsingle{C}})$.\n\\end{rem}\n\n\\subsubsection{Monoidal categories and double categories}\\label{sec:monoidal-cats}\nFor $\\icat{C}=\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$, (non-unital) monoid objects in $\\icat{C}$ are also called \\emph{(non-unital) monoidal $\\infty$-categories}, (non-unital) category objects in $\\icat{C}$ are called \\emph{(non-unital) double $\\infty$-categories}, and (commutative) monoid objects in $\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$ or $\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$ are \\emph{(symmetric) monoidal $\\infty$- or double $\\infty$-categories}. Via the straightening equivalence of \\cref{sec:straightening}, these can be described equivalently as cocartesian fibrations $\\ensuremath{\\catsingle{M}}\\rightarrow\\Delta^{\\mathrm{op}}$ (or $\\ensuremath{\\catsingle{M}}\\rightarrow\\Delta^{\\mathrm{op}}_\\mathrm{inj}$ in the non-unital case, or $\\ensuremath{\\catsingle{M}}\\rightarrow\\ensuremath{\\cat{Fin}}_*$ in the commutative case) such that the functors\n\\[\n\t\\ensuremath{\\catsingle{M}}_{[p]}\\longrightarrow \\ensuremath{\\catsingle{M}}_{[1]}\\times \\ldots \\times \\ensuremath{\\catsingle{M}}_{[1]}\\quad\\text{respectively}\\quad \\ensuremath{\\catsingle{M}}_{[p]}\\longrightarrow \\ensuremath{\\catsingle{M}}_{[1]}\\times_{\\ensuremath{\\catsingle{M}}_{[0]}} \\ldots\\times_{\\ensuremath{\\catsingle{M}}_{[0]}} \\ensuremath{\\catsingle{M}}_{[1]}\n\\]\ninduced by the cocartesian lifts of the Segal maps $\\rho_i$ are equivalences.\n\n\\begin{ex}\\label{ex:cartesian-structure}For an $\\infty$-category $\\ensuremath{\\catsingle{C}}$ with finite products, taking products induces a symmetric monoidal structure $\\ensuremath{\\catsingle{C}}^{\\times}\\rightarrow\\ensuremath{\\cat{Fin}}_*$ on $\\ensuremath{\\catsingle{C}}$, the \\emph{cartesian structure} \\cite[2.4.1]{LurieHA}. Dually, if $\\ensuremath{\\catsingle{C}}$ has finite coproducts, it carries a \\emph{cocartesian} symmetric monoidal structure $\\ensuremath{\\catsingle{C}}^{\\sqcup}\\rightarrow\\ensuremath{\\cat{Fin}}_*$ \\cite[2.4.3]{LurieHA}.\n\\end{ex}\n\n\\begin{rem}\\label{rem:lurie-model-oo-cat}The definition of a monoidal $\\infty$-category given in \\cite[4.1.1.10]{LurieHA} is different from the one given above, but the resulting $\\infty$-categories turn out to be equivalent \\cite[4.1.3]{LurieHA}.\n\\end{rem}\n\n\\subsubsection{Mapping $\\infty$-categories}\\label{sec:mapping-infinity-category}\nGiven a double $\\infty$-category $\\ensuremath{\\catsingle{C}}\\in\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$ and objects $A,B\\in\\ensuremath{\\catsingle{C}}_{[0]}$, we define the \\emph{mapping $\\infty$-category} from $A$ to $B$ to be the $\\infty$-category given as the fibre in $\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$\n\\[\n\t\\gls*{cabmappingcat} \\coloneqq \\mathrm{fib}_{(A,B)}\\big((d_0,d_1)\\colon \\ensuremath{\\catsingle{C}}_{[1]}\\rightarrow \\ensuremath{\\catsingle{C}}_{[0]} \\times \\ensuremath{\\catsingle{C}}_{[0]}\\big).\n\\]\nThese mapping $\\infty$-categories come with composition functors $\\ensuremath{\\catsingle{C}}_{A,B}\\times \\ensuremath{\\catsingle{C}}_{B,C}\\rightarrow\\ensuremath{\\catsingle{C}}_{A,C}$ defined by taking vertical fibres in the commutative diagram in $\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$ \n\\[\\begin{tikzcd}[row sep=0.4cm]\n\t\\ensuremath{\\catsingle{C}}_{[1]} \\times_{\\ensuremath{\\catsingle{C}}[0]} \\ensuremath{\\catsingle{C}}_{[1]} \\dar & \\lar[swap]{\\simeq} \\rar \\ensuremath{\\catsingle{C}}_{[2]} \\dar & \\ensuremath{\\catsingle{C}}_{[1]} \\dar \\\\\n\t\\ensuremath{\\catsingle{C}}_{[0]} \\times \\ensuremath{\\catsingle{C}}_{[0]} \\times \\ensuremath{\\catsingle{C}}_{[0]}  & \\lar[equal] \\ensuremath{\\catsingle{C}}_{[0]} \\times \\ensuremath{\\catsingle{C}}_{[0]} \\times \\ensuremath{\\catsingle{C}}_{[0]} \\rar{\\mathrm{pr}_{1,3}} & \\ensuremath{\\catsingle{C}}_{[0]}  \\times \\ensuremath{\\catsingle{C}}_{[0]}\\end{tikzcd}\\]\nwith top-left horizontal map induced by the Segal morphisms, top-right horizontal map by the unique active morphism $[2] \\to [1]$, and vertical map by the face maps.\n\n\\subsubsection{Quasi-unital monoid and category objects}\\label{sec:quasi-unital}\nA non-unital category object $X\\in\\ensuremath{\\mathrm{Cat}}_{\\mathrm{nu}}(\\ensuremath{\\catsingle{C}})$ is \\emph{quasi-unital} if it admits a \\emph{quasi-unit}, which is by definition a morphism $u\\colon X_{[0]}\\rightarrow X_{[1]}$ together with a commutative diagram in $\\ensuremath{\\catsingle{C}}$\n\\[\\begin{tikzcd}[row sep=0.2cm]\n\tX_{[0]}\\arrow[dr,\"\\mathrm{diag}\",swap]\\arrow[rr,\"u\"]&&X_{[1]}\\arrow[dl,\"{(d_0,d_1)}\"]\\\\[-3pt]\n\t&X_{[0]}\\times X_{[0]}&\n\\end{tikzcd}\\]\nsuch that the following two compositions are equivalent to the identity\n\\begin{equation}\\label{equ:composition-with-qunit}\n\t\\begin{gathered}X_{[1]} \\simeq  X_{[0]}\\times_{X_{[0]}}X_{[1]}\\xrightarrow{(u,\\mathrm{id})}X_{[1]}\\times_{X_{[0]}}X_{[1]}\\simeq X_{[2]}\\xlra{d_1}X_{[1]}\\\\ X_{[1]} \\simeq  X_{[1]}\\times_{X_{[0]}}X_{[0]}\\xrightarrow{(\\mathrm{id},u)}X_{[1]}\\times_{X_{[0]}}X_{[1]} \\simeq  X_{[2]}\\xlra{d_1}X_{[1]}.\\end{gathered}\n\\end{equation}\nQuasi-units are unique up to equivalence \\cite[Remark 4.8]{HaugsengSegal}. A morphism $\\phi\\colon X\\rightarrow Y$ of non-unital category objects is \\emph{quasi-unital} if there exists a commutative diagram in $\\ensuremath{\\catsingle{C}}$ of the form\n\\begin{equation}\\label{equ:morphism-quasiunital}\n\t\\begin{tikzcd}[row sep=0.3cm]\n\t\tX_{[0]}\\arrow[ddr,\"\\mathrm{diag}\", near end,swap]\\arrow[rrr,\"\\phi_0\"]\\arrow[drr,\"u_X\"]&&& Y_{[0]}\\arrow[drr,\"u_Y\"] &&\\\\[-3pt]\n\t\t&& X_{[1]}\\arrow[rrr,\"\\phi_1\",near start]\\arrow[dl,\"{(d_0,d_1)}\",swap]&&& Y_{[1]}\\arrow[dl,\"{(d_0,d_1)}\"]\\\\[-3pt]\n\t\t& X_{[0]}\\times X_{[0]}\\arrow[rrr,\"{(\\phi_0,\\phi_0)}\",swap, near start]&&& Y_{[0]}\\times Y_{[0]}\\arrow[from=uul,\"\\mathrm{diag}\", crossing over, near end, swap]&\n\t\\end{tikzcd}\n\\end{equation}\nsuch that the outer triangles are quasi-units for $X$ and $Y$. As a result of the uniqueness of quasi-units, the composition of two quasi-unital morphisms is quasi-unital. We write $\t\\gls*{catquc} \\subset\\ensuremath{\\mathrm{Cat}}_{\\mathrm{nu}}(\\ensuremath{\\catsingle{C}})$ for the subcategory of \\emph{quasi-unital category objects} in $\\ensuremath{\\catsingle{C}}$, generated by quasi-unital objects and morphisms. Every category object is quasi-unital ($s_0\\colon X_{[0]}\\rightarrow X_{[1]}$ is a quasi-unit), and by \\cite[Theorem 4.14]{HaugsengSegal}, the forgetful functor $\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\catsingle{C}})\\rightarrow\\ensuremath{\\mathrm{Cat}}_{\\mathrm{nu}}(\\ensuremath{\\catsingle{C}})$ induces an equivalence \n\\begin{equation}\\label{equ:qu-is-good}\n\t\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\catsingle{C}})\\overset{\\simeq}\\longrightarrow\\ensuremath{\\mathrm{Cat}}_{\\mathrm{qu}}(\\ensuremath{\\catsingle{C}}).\n\\end{equation}\n\n\\begin{rem}\\label{rem:quasi-unital-into-simplicial} Note that if $X$ a quasi-unital category object in $\\ensuremath{\\catsingle{C}}$, $Y$ a simplicial object in $\\ensuremath{\\catsingle{C}}$ (not necessarily a category object), and $f\\colon X\\rightarrow Y$ a morphism of semisimplicial objects in $\\ensuremath{\\catsingle{C}}$, then\n\t\\begin{enumerate}\n\t\t\\item\\label{enum:quasi-unital-into-simplicial-i}it makes sense to ask for $f$ to be quasi-unital (replace $u_Y$ in \\eqref{equ:morphism-quasiunital} by the $0$th degeneracy map). This property is preserved by postcomposition with maps of simplicial objects,\n\t\t\\item\\label{enum:quasi-unital-into-simplicial-ii} if $\\ensuremath{\\catsingle{C}}=\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$ and $Y'\\subset Y$ is a levelwise full subcategory that is a quasi-unital category object, then a functor $X\\rightarrow Y'$ of non-unital category objects is quasi-unital if and only if the composition $X\\rightarrow Y'\\subset Y$ is quasi-unital in the sense of \\ref{enum:quasi-unital-into-simplicial-i}.\n\t\\end{enumerate}\n\\end{rem}\n\n\\subsubsection{Double $\\infty$-, $(\\infty,2)$-, and $(\\infty,1)$-categories} \\label{sec:double-vs-infty2}\nA double $\\infty$-category has an underlying $(\\infty,2)$-category (in fact two, but we will not need this) which in turn has an underlying $\\infty$-category. More precisely, there are functors of $\\infty$-categories\n\\vspace{-0.1cm}\n\\[\n\t\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}) \\xrightarrow{\\gls*{infty2}} \\ensuremath{\\icat{C}\\mathrm{at}_{(\\infty,2)}} \\xrightarrow{(-)^{\\gls*{infty1}}} \\ensuremath{\\icat{C}\\mathrm{at}_\\infty}.\n\\]\nwhere $\\ensuremath{\\icat{C}\\mathrm{at}_{(\\infty,2)}}$ is the $\\infty$-category of $(\\infty,2)$-categories. We denote the composition by \n\\[\n\t\\gls*{simeq2} \\colon \\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}) \\longrightarrow  \\ensuremath{\\icat{C}\\mathrm{at}_\\infty}.\n\\]\nThese functors have the following properties:\n\\begin{enumerate}\n\\item\\label{enum:oo-2:i} The functors $(-)^{(\\infty,2)}$ and $(-)^{\\simeq_2}$ preserve finite products and hence (symmetric) monoidal structures, and so does their composition $(-)^{(\\infty,1)}$.\n\\item \\label{enum:oo-2:ii}For $\\ensuremath{\\catsingle{C}}\\in\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$, the objects of $\\ensuremath{\\catsingle{C}}^{(\\infty,2)}$ can be identified with those of $\\ensuremath{\\catsingle{C}}$. The analogous property holds for the functor $(-)^{\\simeq_2}$ and thus also for their composition $(-)^{(\\infty,1)}$.\n\\item\\label{enum:oo-2:iii} For $\\ensuremath{\\catsingle{C}}\\in\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$, the mapping $\\infty$-category $\\ensuremath{\\catsingle{C}}_{A,B}$ between two objects $A$ and $B$ in $\\ensuremath{\\catsingle{C}}$ can be identified with the corresponding mapping $\\infty$-category in $\\ensuremath{\\catsingle{C}}^{(\\infty,2)}$. The functor $(-)^{\\simeq_2}$ is on mapping $\\infty$-categories given by taking cores (hence the notation), and thus the same holds for $(-)^{(\\infty,1)}$, so we have $\\ensuremath{\\catsingle{C}}_{A,B}^\\simeq\\simeq\\mathrm{Map}_{\\ensuremath{\\catsingle{C}}^{(\\infty,1)}}(A,B)$ for objects $A$ and $B$ in $\\ensuremath{\\catsingle{C}}$.\n\\end{enumerate}\nOne way to implement these $\\infty$-categories and functors between them is to use the equivalence $\\ensuremath{\\mathrm{Cat}}_\\infty\\simeq \\ensuremath{\\icat{C}\\mathrm{SS}}(\\ensuremath{\\catsingle{S}})$ to Rezk's \\emph{complete Segal spaces} (a certain full subcategory of $\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\catsingle{S}})$ \\cite[Section 3]{HaugsengSpans}) and model $\\ensuremath{\\icat{C}\\mathrm{at}_{(\\infty,2)}}$ as the $\\infty$-category of \\emph{$2$-fold complete Segal spaces} $\\ensuremath{\\icat{C}\\mathrm{SS}}_2(\\ensuremath{\\catsingle{S}})$ in the sense of Barwick (a certain full subcategory of $\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\catsingle{S}}))$ \\cite[Section 4]{HaugsengSpans}). In these models, the functor $(-)^{(\\infty,2)}$ is explained in \\cite[Remark 3.15]{HaugsengMorita} and the functor $(-)^{(\\infty,1)}$ can be constructed via the inductive description as 2-fold Segal spaces as $\\ensuremath{\\icat{C}\\mathrm{SS}}_2(\\ensuremath{\\catsingle{S}})=\\ensuremath{\\icat{C}\\mathrm{SS}}_{\\ensuremath{\\icat{C}\\mathrm{SS}}(\\ensuremath{\\catsingle{S}})}(\\ensuremath{\\icat{C}\\mathrm{SS}}(\\ensuremath{\\catsingle{S}}))$ \\cite[Section 7]{HaugsengSpans} by defining $(-)^{\\simeq_2}$ as the right-adjoint $\\ensuremath{\\icat{C}\\mathrm{SS}}_{\\ensuremath{\\icat{C}\\mathrm{SS}}(\\ensuremath{\\catsingle{S}})}(\\ensuremath{\\icat{C}\\mathrm{SS}}(\\ensuremath{\\catsingle{S}}))\\rightarrow \\ensuremath{\\icat{C}\\mathrm{SS}}_\\ensuremath{\\catsingle{S}}(\\ensuremath{\\catsingle{S}})=\\ensuremath{\\icat{C}\\mathrm{SS}}(\\ensuremath{\\catsingle{S}})\\simeq \\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$ induced by the right-adjoint $\\mathrm{ev}_{[0]}\\colon \\ensuremath{\\icat{C}\\mathrm{SS}}(\\ensuremath{\\catsingle{S}})\\rightarrow \\ensuremath{\\catsingle{S}}$ to the inclusion $c\\colon \\ensuremath{\\catsingle{S}}\\rightarrow \\ensuremath{\\icat{C}\\mathrm{SS}}(\\ensuremath{\\catsingle{S}})$ as constant simplicial spaces, using \\cite[Proposition 7.17]{HaugsengSpans}.\n\nIt remains to justify properties \\ref{enum:oo-2:i}--\\ref{enum:oo-2:iii}. That  \\ref{enum:oo-2:i} holds for $(-)^{(\\infty,2)}$ is justified in \\cite[Remark 3.15]{HaugsengMorita} and for $(-)^{\\simeq_2}$ it holds since it is a right adjoint. For \\ref{enum:oo-2:ii} and \\ref{enum:oo-2:iii}, one uses \\cite[Lemma 5.51]{HaugsengMorita} and that $\\mathrm{ev}_{[0]}$ corresponds to taking cores under the equivalence $\\ensuremath{\\icat{C}\\mathrm{at}_\\infty} \\simeq \\ensuremath{\\icat{C}\\mathrm{SS}}(\\ensuremath{\\catsingle{S}})$. \n\n\\subsection{Presheaves and the Yoneda embedding}\\label{sec:presheaf-category}\\label{sec:yoneda-day}\nGiven an $\\infty$-category $\\ensuremath{\\catsingle{C}}$, we write $\\gls*{psh} \\coloneqq \\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\catsingle{C}}^{\\mathrm{op}},\\ensuremath{\\catsingle{S}})$ for the $\\infty$-category of $\\ensuremath{\\catsingle{S}}$-valued presheaves on $\\ensuremath{\\catsingle{C}}$. This admits all small limits and colimits  \\cite[5.1.2.4]{LurieHTT}, and there is a natural fully faithful \\emph{Yoneda embedding} \\cite[5.1.3.1]{LurieHTT}\n\\[\n\t\\gls*{yon} \\colon \\ensuremath{\\catsingle{C}}\\lhook\\joinrel\\longrightarrow \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\catsingle{C}}).\n\\]\n If $\\ensuremath{\\catsingle{C}}$ is (symmetric) monoidal, then its opposite $\\ensuremath{\\catsingle{C}}^{\\mathrm{op}}$ is (symmetric) monoidal \\cite[2.4.2.7]{LurieHA}, and $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\catsingle{C}})$ carries a (symmetric) monoidal structure by Day convolution \\cite[2.2.6.17]{LurieHA} with respect to which the Yoneda embedding can be enhanced to a (symmetric) monoidal functor preserving small colimits in each variable \\cite[4.8.1.12, 4.8.1.13]{LurieHA}. Explicitly, a formula for Day convolution is given by\n$(F \\otimes G)(c'') = \\mathrm{colim}_{c'' \\to c \\otimes c'} (F(c) \\times G(c'))$\nwhere the colimit is over the category of triples $(c,c',u)$ with $c,c' \\in \\ensuremath{\\catsingle{C}}$ and $u \\colon c'' \\to c \\otimes c'$ \\cite[2.2.6]{LurieHA}. In particular, as sifted colimits commute with finite products by \\cite[5.5.8.11]{LurieHTT}, Day convolution preserves sifted colimits (such as geometric realisations) in both variables. Moreover, from the construction, one sees that a lax (symmetric) monoidal functor $\\ensuremath{\\catsingle{C}}\\rightarrow \\ensuremath{\\catsingle{D}}$ (see \\cref{ex:monoidal-cat-as-operads})  induces a lax (symmetric) monoidal functor $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\catsingle{D}})\\rightarrow\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\catsingle{C}})$ by precomposition.\n\n\\begin{rem}\\label{fact:yoneda-comparison}Given a $\\ensuremath{\\cat{Kan}}$-enriched category $\\ensuremath{\\cat{C}}$, there is a similar Yoneda embedding in $\\ensuremath{\\cat{Kan}}$-enriched categories $y_s\\colon \\ensuremath{\\cat{C}}\\rightarrow\\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{C}}^{\\mathrm{op}},\\ensuremath{\\cat{Kan}})$. Taking coherent nerves and postcomposing with the map $N_{\\mathrm{coh}}(\\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{C}}^{\\mathrm{op}},\\ensuremath{\\cat{Kan}}))\\rightarrow \\ensuremath{\\mathrm{Fun}}(N_{\\mathrm{coh}}(\\ensuremath{\\cat{C}})^{\\mathrm{op}},N_{\\mathrm{coh}}(\\ensuremath{\\cat{Kan}}))=\\ensuremath{\\mathrm{PSh}}(N_{\\mathrm{coh}}(\\ensuremath{\\cat{C}}))$ of \\cref{sec:coherent-nerve-props} \\ref{enum:coherent-functor} yields a functor $N_{\\mathrm{coh}}(\\ensuremath{\\cat{C}})\\rightarrow \\ensuremath{\\mathrm{PSh}}(N_{\\mathrm{coh}}(\\ensuremath{\\cat{C}}))$ which turns out to agree with $y$ up to equivalence, by the construction of $y$ for $\\ensuremath{\\catsingle{C}}=N_{\\mathrm{coh}}(\\ensuremath{\\catsingle{C}})$ in \\cite[5.1.3.1]{LurieHTT}.\\end{rem}\n\n\\subsection{$\\infty$-operads}\\label{sec:gen-infty-operads} \\label{sec:infty-operads} \nRecall the following definition from \\cite[2.1.1.10]{LurieHA}.\n\\begin{dfn} A \\emph{$\\infty$-operad} $\\ensuremath{\\catsingle{O}}$ is a functor $p \\colon \\ensuremath{\\catsingle{O}}^\\otimes \\to \\ensuremath{\\cat{Fin}}_\\ast$ that satisfies:\n\t\\begin{enumerate}\n\t\t\\item $\\ensuremath{\\catsingle{O}}^\\otimes$ has cocartesian lifts for inert morphisms in $\\ensuremath{\\cat{Fin}}_\\ast$,\n\t\t\\item\\label{enum:operad-ii} the map\n\t\t$\\sqcap (\\rho_i)_!\\colon\\ensuremath{\\catsingle{O}}^\\otimes_{\\langle n \\rangle} \\rightarrow \\sqcap_{i=1}^n  \\ensuremath{\\catsingle{O}}^\\otimes_{\\langle 1 \\rangle}$\n\t\tinduced by the Segal morphisms is an equivalence,\n\t\t\\item given an object $x \\in \\ensuremath{\\catsingle{O}}^\\otimes_{\\langle n \\rangle}$ and cocartesian lifts $x \\to x_i$ of the Segal morphisms $\\rho_i \\colon \\langle n \\rangle \\to \\langle 1 \\rangle$, the  following commutative diagram in $\\ensuremath{\\catsingle{S}}$ is cartesian\n\t\t\\[\\begin{tikzcd} \\mathrm{Map}_\\ensuremath{\\catsingle{O}}(y,x) \\rar \\dar & \\sqcap_{i=1}^n \\mathrm{Map}_\\ensuremath{\\catsingle{O}}(y,x_i) \\dar \\\\[-5pt]\n\t\t\t\\mathrm{Map}_{\\ensuremath{\\cat{Fin}}_\\ast}(\\langle m \\rangle,\\langle n \\rangle) \\rar & \\sqcap_{i=1}^n \\mathrm{Map}_{\\ensuremath{\\cat{Fin}}_\\ast}(\\langle m \\rangle,\\langle 1 \\rangle).\\end{tikzcd}\\]\n\t\\end{enumerate}\n\t\\end{dfn}\n\tA \\emph{map of $\\infty$-operads} is a functor over $\\ensuremath{\\cat{Fin}}_\\ast$ that preserves cocartesian lifts over inert morphisms. Such a map $\\ensuremath{\\catsingle{O}}^{\\otimes} \\to \\ensuremath{\\catsingle{P}}^{\\otimes}$ is also called an \\emph{$\\ensuremath{\\catsingle{O}}$-algebra in $\\ensuremath{\\catsingle{P}}$}, and we write \n\\[\\ensuremath{\\mathrm{Alg}}_{\\ensuremath{\\catsingle{O}}}(\\ensuremath{\\catsingle{P}})\\subset \\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Fin}}_*}(\\ensuremath{\\catsingle{O}}^{\\otimes},\\ensuremath{\\catsingle{P}}^{\\otimes})\\] \nfor the full subcategory of such maps. Given an $\\infty$-operad $\\ensuremath{\\catsingle{O}}$, we call the objects of $\\ensuremath{\\catsingle{O}}^\\otimes_{\\langle 1 \\rangle}$  the \\emph{colours of $\\ensuremath{\\catsingle{O}}$}. Given colours $(x_1,\\ldots,x_n) \\in \\sqcap^n \\smash{\\ensuremath{\\catsingle{O}}^\\otimes_{\\langle 1 \\rangle}}\\simeq \\smash{\\ensuremath{\\catsingle{O}}^\\otimes_{\\langle n \\rangle}}$ and $y \\in \\smash{\\ensuremath{\\catsingle{O}}^\\otimes_{\\langle 1 \\rangle}}$, the  \\emph{space of multi-operations} is the subspace $\\gls*{mul}(x,y) \\subset \\mathrm{Map}_{\\ensuremath{\\catsingle{O}}^\\otimes}(x,y)$ covering the unique active morphism $\\langle n \\rangle \\to \\langle 1 \\rangle$ \\cite[2.1.1.16]{LurieHA}. If $\\ensuremath{\\catsingle{O}}^\\otimes$ has a single colour $x\\in \\ensuremath{\\catsingle{O}}^\\otimes_{\\langle 1 \\rangle}$, we abbreviate $\\ensuremath{\\catsingle{O}}(k)\\coloneq \\mathrm{Mul}_\\ensuremath{\\catsingle{O}}(x,\\ldots,x;x)$ where $x$ appears in the domain $k$ times. These spaces of multi-operations can be composed using \\emph{operadic composition maps}, denoted $\\gls*{circo}$, that satisfy the axioms of a coloured operad in the classical sense up to coherent homotopies \\cite[2.1.1.17]{LurieHA}. In particular, the \\emph{homotopy operad} $h\\ensuremath{\\catsingle{O}}^\\otimes\\rightarrow \\ensuremath{\\cat{Fin}}_*$ (which is an operad as a result of the properties of $h$ discussed in \\cref{sec:scat-vs-qcat} and satisfies $\\mathrm{Mul}_{h\\ensuremath{\\catsingle{O}}}(x,y)=\\pi_0\\,\\mathrm{Mul}_{\\ensuremath{\\catsingle{O}}}(x,y)$) gives a coloured operad in the classical sense. By construction, there is a map of $\\infty$-operads $\\ensuremath{\\catsingle{O}}^\\otimes\\rightarrow h\\ensuremath{\\catsingle{O}}^\\otimes$. \n\n\\begin{ex}\\label{ex:monoidal-cat-as-operads}When viewed as a cocartesian fibration $\\ensuremath{\\catsingle{C}}^{\\otimes}\\rightarrow\\ensuremath{\\cat{Fin}}_\\ast$ (see \\cref{sec:monoidal-cats}), every symmetric monoidal category $\\ensuremath{\\catsingle{C}}$ is an $\\infty$-operad. A map of $\\infty$-operads between symmetric monoidal categories is called a \\emph{lax symmetric monoidal functor}.\n\\end{ex}\n\n\\begin{ex}\\label{ex:associative-operad}\nEvery coloured operad in the category of $\\ensuremath{\\cat{Kan}}$-complexes in the classical sense gives rise to an $\\infty$-operad via the \\emph{operadic nerve} \\cite[2.1.1.27]{LurieHTT}. For example, the \\emph{associative $\\infty$-operad} $\\ensuremath{\\icat{A}\\mathrm{ssoc}}$ \\cite[4.1.1.1, 4.1.1.3]{LurieHA} is the operadic nerve of the ordinary operad with a single colour $\\ast$, whose $k$-ary multi-operations $\\ensuremath{\\icat{A}\\mathrm{ssoc}}(k) = \\mathrm{Mul}_{\\ensuremath{\\icat{A}\\mathrm{ssoc}}}(*,\\ldots,*;*)$ is the set of linear orders of $\\ul{k} = \\{1,2,\\ldots,k\\}$, and where operadic composition is concatenation of linear orders. An $\\infty$-operad $\\ensuremath{\\catsingle{O}}$ is equivalent to $\\ensuremath{\\icat{A}\\mathrm{ssoc}}$ if and only if there is an isomorphism $h\\ensuremath{\\catsingle{O}} \\cong h\\ensuremath{\\icat{A}\\mathrm{ssoc}}$ of operads in the $1$-category of sets and all spaces of operations in $\\ensuremath{\\catsingle{O}}$ are homotopy discrete.\n\\end{ex}\n\n\\subsubsection{Suboperads, endomorphism operads, and algebras over them}\\label{sec:suboperad}\\label{sec:end-operads} \\label{sec:map-as-algebra} \nLet $\\ensuremath{\\catsingle{O}}$ be an $\\infty$-operad and $\\cat{O}_0 \\subseteq h\\ensuremath{\\catsingle{O}}$ be a suboperad of the ordinary operad $h\\ensuremath{\\catsingle{O}}$ in sets. The corresponding \\emph{suboperad} $\\ensuremath{\\catsingle{O}} \\times_{h \\ensuremath{\\catsingle{O}}} \\cat{O}_0$ of $\\ensuremath{\\catsingle{O}}$ is defined as the pullback $\\ensuremath{\\catsingle{O}}^\\otimes \\times_{h\\ensuremath{\\catsingle{O}}^\\otimes} \\cat{O}_0^\\otimes\\rightarrow \\ensuremath{\\cat{Fin}}_\\ast$ in the $\\infty$-category $\\ensuremath{{\\icat{O}\\mathrm{pd}_\\infty}}$ of $\\infty$-operads, which has limits by \\cite[2.1.4]{LurieHA}. In particular, we may restrict $\\ensuremath{\\catsingle{O}}$ to a fixed collection of colours closed under equivalences to obtain a new $\\infty$-operad. We call this a \\emph{full suboperad}. \n\n\\begin{rem}\nThe forgetful functor $\\ensuremath{{\\icat{O}\\mathrm{pd}_\\infty}} \\to (\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})_{\/\\ensuremath{\\cat{Fin}}_\\ast}$ creates limits by \\cite[Lemma 1.13]{AyalaFrancisTanakaFact}, so the underlying $\\infty$-category of $\\ensuremath{\\catsingle{O}} \\times_{h \\ensuremath{\\catsingle{O}}} \\cat{O}_0$ agrees with the analogous pullback in $\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$.\n\\end{rem}\n\n\\begin{ex}\\label{ex:sub-sym-monoidal}\nFor a symmetric $\\infty$-monoidal category $\\ensuremath{\\catsingle{C}}$ viewed as an $\\infty$-operad, its homotopy operad $h\\ensuremath{\\catsingle{C}}$ is a symmetric monoidal $1$-category in the classical sense. Given a sub symmetric monoidal category of $\\cat{C}_0\\subset h\\ensuremath{\\catsingle{C}}$ in the $1$-categorical sense, the associated sub $\\infty$-operad $\\ensuremath{\\catsingle{C}} \\times_{h \\ensuremath{\\catsingle{C}}} \\cat{C}_0$ is again a symmetric $\\infty$-monoidal category. Informally, this is given by restricting the objects and the components of the mapping spaces according to $\\cat{C}_0$.\n\\end{ex}\n\nFix $\\ensuremath{\\catsingle{C}}$ a symmetric monoidal $\\infty$-category $\\ensuremath{\\catsingle{C}}$,  viewed as an $\\infty$-operad. The \\emph{endomorphism operad} of an object $x$ in $\\ensuremath{\\catsingle{C}}$ is the full sub $\\infty$-operad \\gls*{endc} obtained by restricting to the colours equivalent to $x$. Writing $\\mathbbm{1}$ for the unit in $\\ensuremath{\\catsingle{C}}$, we can form the composition of maps of $\\infty$-operads\n\\begin{equation}\\label{equ:algebra-over-end}\\mathrm{End}_\\ensuremath{\\catsingle{C}}(x)^{\\otimes} \\xra{\\subset} \\ensuremath{\\catsingle{C}}^\\otimes \\overset{y}\\longrightarrow \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\catsingle{C}})^{\\otimes} \\overset{\\mathrm{ev}_\\mathbbm{1}}\\longrightarrow \\ensuremath{\\catsingle{S}}^\\times\\end{equation}\nto $\\ensuremath{\\catsingle{S}}$ equipped with the cartesian symmetric monoidal structure (see \\cref{ex:cartesian-structure}). The first map is induced by the inclusion, the second map the symmetric monoidal Yoneda embedding (see \\cref{sec:yoneda-day}), and the third map the evaluation at the unit which is a map of $\\infty$-operads by naturality of the Day convolution in lax symmetric monoidal functors (see \\cref{sec:yoneda-day}). The composition \\eqref{equ:algebra-over-end}  enhances the mapping space $\\mathrm{Map}_\\ensuremath{\\catsingle{C}}(\\mathbbm{1},x)$ to an $\\mathrm{End}_\\ensuremath{\\catsingle{C}}(x)$-algebra in $\\ensuremath{\\catsingle{S}}$.\n\n\\subsubsection{Generalised $\\infty$-operads}The condition \\ref{enum:operad-ii} in the definition of an $\\infty$-operad $\\ensuremath{\\catsingle{O}}$ in particular implies that $\\smash{\\ensuremath{\\catsingle{O}}^{\\otimes}_{\\langle0\\rangle}}$ is trivial. Sometimes it it useful to relax the notion of an $\\infty$-operad to that of a \\emph{generalised $\\infty$-operad} which need no longer satisfy $\\smash{\\ensuremath{\\catsingle{O}}^{\\otimes}_{\\langle0\\rangle}}\\simeq\\ast$. The precise definition of a generalised $\\infty$-operad is not important for us, but it suffices to know that it is a functor $\\ensuremath{\\catsingle{O}}^{\\otimes}\\rightarrow \\ensuremath{\\cat{Fin}}_\\ast$ satisfying some weaker axioms than those for $\\infty$-operads, but that the existence of cocartesian lifts for inert morphisms is still required. \\emph{Maps of generalised operads} $\\ensuremath{\\catsingle{O}}\\rightarrow\\ensuremath{\\catsingle{P}}$ are defined in the same way as for $\\infty$-operads. Generalising the case of $\\infty$-operads, we denote the resulting subcategory by $\\ensuremath{\\mathrm{Alg}}_{\\ensuremath{\\catsingle{O}}}(\\ensuremath{\\catsingle{P}})\\subset\\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Fin}}_\\ast}(\\ensuremath{\\catsingle{O}}^{\\otimes},\\ensuremath{\\catsingle{P}}^{\\otimes})$ and still call its objects $\\ensuremath{\\catsingle{O}}$-algebras in $\\ensuremath{\\catsingle{P}}$.\n\n\\subsubsection{(Generalised) nonsymmetric $\\infty$-operads}\\label{sec:gen-operads}\nReplacing the category $\\ensuremath{\\cat{Fin}}_*$ by $\\Delta^\\mathrm{op}$ defines \\emph{nonsymmetric} variants of all of the above definitions and constructions, e.g.\\,(generalised) nonsymmetric operads, maps between them, algebras in them, etc. We use the same notation for the symmetric and nonsymmetric constructions, e.g.\\,for (generalised) nonsymmetric $\\infty$-operads $\\ensuremath{\\catsingle{O}}$ and $\\ensuremath{\\catsingle{P}}$, we write $\\ensuremath{\\mathrm{Alg}}_{\\ensuremath{\\catsingle{O}}}(\\ensuremath{\\catsingle{P}})\\subset \\ensuremath{\\mathrm{Fun}}_{\\Delta^{\\mathrm{op}}}(\\ensuremath{\\catsingle{O}}^{\\otimes},\\ensuremath{\\catsingle{P}}^{\\otimes})$ for the $\\infty$-category of maps of (generalised) nonsymmetric $\\infty$-operads aka $\\ensuremath{\\catsingle{O}}$-algebras in $\\ensuremath{\\catsingle{P}}$. \n\n\\begin{ex}\\label{example:gen-ns-operad}\\label{example:maps-gen-ns-operad}\n\n\tThe following examples of generalised nonsymmetric $\\infty$-operads will be important:\n\t\\begin{enumerate}\n\t\t\\item Cocartesian fibrations obtained by unstraightening double $\\infty$-categories.\n\t\t\\item The projection $\\Delta^{\\mathrm{op}}_{\/[p]}\\rightarrow \\Delta^{\\mathrm{op}}$ for all $p\\ge0$, see \\cite[Lemma 4.10]{HaugsengMorita}.\n\t\t\\item The restriction $\\Lambda^{\\mathrm{op}}_{\/[p]}\\rightarrow \\Delta^{\\mathrm{op}}$ of the projection $\\Delta^{\\mathrm{op}}_{\/[p]}\\rightarrow \\Delta^{\\mathrm{op}}$ to the full subcategory $\\gls*{lambda}\\subset \\Delta_{\/[p]}$ spanned by the cellular maps in $\\Delta$, see \\cite[Lemma 4.14]{HaugsengMorita}.\n\t\\end{enumerate}\n\tExamples of maps between generalised nonsymmetric $\\infty$-operads that will be important are:\n\t\\begin{enumerate}\n\t\t\\item The map $\\Delta^\\mathrm{op}_{\/[p]} \\to \\Delta^\\mathrm{op}_{\/[q]}$ over $\\Delta^\\mathrm{op}$ induced by a morphism $[p] \\to [q]$ of $\\Delta$.\n\t\t\\item The inclusion $\\Lambda^\\mathrm{op}_{\/[p]} \\to \\Delta^\\mathrm{op}_{[p]}$ over $\\Delta^\\mathrm{op}$ \\cite[Lemma 4.14]{HaugsengMorita}.\n\t\\end{enumerate}\n\\end{ex}\n\n\\subsection{Associative algebras and bimodules in the nonsymmetric setting}\\label{sec:assalg-bimodules}\nGiven a monoidal $\\infty$-category viewed as a cocartesian fibration $\\ensuremath{\\catsingle{C}}^{\\otimes}\\rightarrow \\Delta^{\\mathrm{op}}$ with underlying category $\\ensuremath{\\catsingle{C}}\\coloneqq \\ensuremath{\\catsingle{C}}^{\\otimes}_{[1]}$, the $\\infty$-categories $\\gls*{assc}$ and $\\gls*{bimodc}$ of \\emph{associative algebras in $\\ensuremath{\\catsingle{C}}$} and \\emph{bimodules in $\\ensuremath{\\catsingle{C}}$} are defined as\n\\[\n\t\\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})\\coloneqq \\ensuremath{\\mathrm{Alg}}_{\\Delta^{\\mathrm{op}}}(\\ensuremath{\\catsingle{C}}^{\\otimes})\n\t\\quad\\text{and}\\quad\n\t\\ensuremath{\\mathrm{Bimod}}(\\ensuremath{\\catsingle{C}})\\coloneqq \\ensuremath{\\mathrm{Alg}}_{\\Delta^{\\mathrm{op}}_{\/[1]}}(\\ensuremath{\\catsingle{C}}^{\\otimes}).\n\\]\nThese are the $\\infty$-categories of $\\Delta^{\\mathrm{op}}$- and $\\Delta_{\/[1]}^{\\mathrm{op}}$-algebras in $\\ensuremath{\\catsingle{C}}$ as in \\cref{sec:gen-operads}. There is a functor\n\\begin{equation}\\label{equ:bimodule-underlying-objects}\\ensuremath{\\mathrm{Bimod}}(\\ensuremath{\\catsingle{C}})\\longrightarrow \\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})\\times \\ensuremath{\\catsingle{C}}\\times \\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})\\end{equation} \nconsisting of the projections to $\\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})$ induced by precomposition with the functors $ \\Delta=\\Delta_{\/[0]}\\rightarrow \\Delta_{\/[1]}$ induced by the $0$th and $1$st face map $[0]\\rightarrow[1]$, and the functor to $\\smash{\\ensuremath{\\catsingle{C}}^{\\otimes}_{[1]}}=\\ensuremath{\\catsingle{C}}$ given by evaluation at $\\mathrm{id}_{[1]}\\in\\Delta_{\/[1]}$. The fibre in $\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$\n\\[\\ensuremath{\\mathrm{Bimod}}_{A,B}(\\ensuremath{\\catsingle{C}})\\coloneqq \\mathrm{fib}_{(A,B)}\\big(\\ensuremath{\\mathrm{Bimod}}(\\ensuremath{\\catsingle{C}})\\rightarrow \\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})\\times \\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})\\big)\\]\nat $(A,B)$ for associative algebras $A,B\\in \\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})$ of the postcomposition $\\ensuremath{\\mathrm{Bimod}}(\\ensuremath{\\catsingle{C}})\\rightarrow \\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})\\times \\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})$ of \\eqref{equ:bimodule-underlying-objects} with the projection is the \\emph{$\\infty$-category of $(A,B)$-bimodules}.\n\n\\begin{rem}\\label{fact:ass-vs-mon}\nAssociative algebras are closely related to monoid objects in the sense of \\cref{sec:cat-objects}: for a category $\\ensuremath{\\catsingle{C}}$ with finite products, equipped with the cartesian monoidal structure (see \\cref{ex:cartesian-structure}), we have an equivalence of $\\infty$-categories $\\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}}^\\times)\\simeq\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{C}})$ \\cite[2.4.2.5]{LurieHA}.\n\\end{rem}\n\n\\begin{rem}\\label{rem:lurie-bimodules} Lurie uses different models for the $\\infty$-categories of associative algebras and bimodules in a monoidal $\\infty$-category $\\ensuremath{\\catsingle{C}}$ (using the equivalent point of view on monoidal structures mentioned in \\cref{rem:lurie-model-oo-cat}), but these turn out to be equivalent to $\\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})$ and $\\ensuremath{\\mathrm{Bimod}}(\\ensuremath{\\catsingle{C}})$ as defined above. For $\\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})$ this is proved as \\cite[4.1.3.19]{LurieHA} and for $\\ensuremath{\\mathrm{Bimod}}(\\ensuremath{\\catsingle{C}})$ it follows from an extension of that argument, but we will not need this comparison in this work.\n\\end{rem}\nThe following lemma of Haugseng on free $(A,B)$-bimodules will be important later:\n\n\\begin{lem}\\label{lemma:free-modules}For a monoidal $\\infty$-category $\\ensuremath{\\catsingle{C}}$ and associative algebras $A,B\\in\\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})$, the composition $\\gls*{uab}\\colon \\ensuremath{\\mathrm{Bimod}}_{A,B}(\\ensuremath{\\catsingle{C}})\\rightarrow \\ensuremath{\\catsingle{C}}$ of the inclusion into $\\ensuremath{\\mathrm{Bimod}}(\\ensuremath{\\catsingle{C}})$ followed by \\eqref{equ:bimodule-underlying-objects} and the projection to $\\ensuremath{\\catsingle{C}}$ has the following properties:\n\t\\begin{enumerate}\n\t\t\\item\\label{enum:free-modules-ii} It detects colimits indexed by weakly contractible simplicial sets.\n\t\t\\item\\label{enum:free-modules-iii} It reflects equivalences.\n\t\t\\item\\label{enum:free-modules-i} It has a left-adjoint $\\gls*{fab}\\colon \\ensuremath{\\catsingle{C}}\\rightarrow \\ensuremath{\\mathrm{Bimod}}_{A,B}(\\ensuremath{\\catsingle{C}})$ whose unit $M\\rightarrow U_{A,B}F_{A,B}(M)$ for $M\\in \\ensuremath{\\catsingle{C}}$ agrees with the map $M\\rightarrow A\\otimes M\\otimes B$ given by tensoring with the units of $A$ and $B$.\n\t\t\\item\\label{enum:free-modules-iv} For a functor $\\varphi\\colon \\ensuremath{\\catsingle{C}}\\rightarrow\\ensuremath{\\catsingle{D}}$ of monoidal $\\infty$-categories and $M\\in\\ensuremath{\\catsingle{C}}$, the canonical morphism $F_{\\varphi(A),\\varphi(B)}(\\varphi(M))\\rightarrow \\varphi(F_{A,B}(M))$ is an equivalence.\n\t\\end{enumerate}\n\\end{lem}\n\\begin{proof}This is a consequence of \\cite[Corollary 4.49]{HaugsengMorita}: the second part of this corollary shows \\ref{enum:free-modules-ii} for diagrams indexed by sifted simplicial sets, but the given proof goes through more generally for weakly contractible simplicial sets. The final part of this corollary in particular shows \\ref{enum:free-modules-iii} since right adjoints in monadic adjunctions reflect equivalences \\cite[4.7.3.5]{LurieHA} and the first part shows \\ref{enum:free-modules-i}. This leaves \\ref{enum:free-modules-iv}. As a result of \\ref{enum:free-modules-iii} it suffices to show that \n\t\\[U_{\\varphi(A),\\varphi(B)}F_{\\varphi(A),\\varphi(B)}(\\varphi(M))\\longrightarrow U_{\\varphi(A),\\varphi(B)}(\\varphi(F_{A,B}(M)))\\simeq\\varphi(U_{A,B}F_{A,B}(M))\\] is an equivalence. Using the second part of \\ref{enum:free-modules-i} this follows from the monoidality of $\\varphi$.\n\\end{proof}\n\n\\subsection{Haugseng's Morita category}\\label{sec:haugseng-morita} In analogy with the classical Morita category of a ring, for a sufficiently nice monoidal $\\infty$-category $\\ensuremath{\\catsingle{C}}$ one would expect a double $\\infty$-category $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})$---the \\emph{Morita category} of $\\ensuremath{\\catsingle{C}}$---whose $\\infty$-category of objects $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})_{[0]}$ is the $\\infty$-category of associative algebras $\\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})$, whose $\\infty$-category of morphisms $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})_{[1]}$ is the category of bimodules $\\ensuremath{\\mathrm{Bimod}}(\\ensuremath{\\catsingle{C}})$, and whose composition is given by ``tensoring bimodules''. Haugseng constructed such a Morita category in \\cite{HaugsengMorita} (denoted $\\ensuremath{\\mathrm{ALG}}_1(\\ensuremath{\\catsingle{C}})$ therein) under mild assumptions on $\\ensuremath{\\catsingle{C}}$. In what follows, we recall his construction and establish some properties not explicitly stated in \\cite{HaugsengMorita}.\n\n\\subsubsection{The pre-Morita category}\\label{sec:pre-morita}\nFor a monoidal $\\infty$-category $\\ensuremath{\\catsingle{C}}^{\\otimes}\\rightarrow \\Delta^{\\mathrm{op}}$, the \\emph{pre-Morita simplicial $\\infty$-category of $\\ensuremath{\\catsingle{C}}$} is the simplicial $\\infty$-category $\\gls*{prealg}\\in\\ensuremath{\\mathrm{Fun}}(\\Delta^\\mathrm{op},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$ with\n\\[\n\t\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}} )_{[p]}\\coloneqq \\ensuremath{\\mathrm{Alg}}_{\\Delta^{\\mathrm{op}}_{\/[p]}}(\\ensuremath{\\catsingle{C}}^{\\otimes})\\subset \\ensuremath{\\mathrm{Fun}}_{\\Delta^{\\mathrm{op}}}(\\Delta^{\\mathrm{op}}_{\/[p]},\\ensuremath{\\catsingle{C}}^{\\otimes}).\n\\] \nThe simplicial structure is given by precomposition with the functors $\\smash{\\Delta_{\/[p]}\\rightarrow \\Delta_{\/[q]}}$ induced by postcomposition with morphisms $[p]\\rightarrow [q]$ in $\\Delta$; this uses \\cref{example:maps-gen-ns-operad}. By construction, $\\smash{\\overline{\\ensuremath{\\mathrm{ALG}}}(-)}$ is natural in lax monoidal functors by postcomposition.\n\nThis definition extends the $\\infty$-categories $\\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})=\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}} )_{[0]}$ and $\\ensuremath{\\mathrm{Bimod}}(\\ensuremath{\\catsingle{C}})=\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}} )_{[1]}$ to a simplicial $\\infty$-category $\\smash{\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}})}$, but the result is not yet a double $\\infty$-category: an object in $\\smash{\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}})_[p]}$ gives associative algebras $M(i)$ for $0 \\leq i \\leq p$ and $(M(i),M(j))$-bimodules $M(i,j)$ for $0 \\leq i<j \\leq p$ and, informally speaking, we need to enforce that $M(i,j)$ is equivalent to the iterated tensor product $M(i,i+1) \\otimes_{M(i+1)} M(i+1,i+2) \\otimes_{M(i+2)} \\cdots \\otimes_{M(j-1)} M(j-1,j)$.\n\n\\subsubsection{Composite algebras and the Morita category}\\label{sec:composite-algebras}\nThe condition just mentioned can be made precise through the notion of a \\emph{composite algebra} from \\cite[Section 4.2]{HaugsengMorita}. It requires an assumption on $\\ensuremath{\\catsingle{C}}$ that Haugseng calls \\emph{having good relative tensor products} \\cite[Definition 4.18]{HaugsengMorita}, which is in particular satisfied if the underlying category $\\ensuremath{\\catsingle{C}}$ admits all geometric realisations (colimits indexed over $\\Delta^{\\mathrm{op}}$) and if these geometric realisations are preserved by tensoring (on either side) with fixed objects of $\\ensuremath{\\catsingle{C}}$; this follows from \\cite[Lemma 4.19]{HaugsengMorita}. If $\\ensuremath{\\catsingle{C}}$ has good relative tensor products, then the functor $\\smash{\\tau_p^* \\colon \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}} )_{[p]}=\\ensuremath{\\mathrm{Alg}}_{\\Delta^{\\mathrm{op}}_{\/[p]}}(\\ensuremath{\\catsingle{C}}^{\\otimes})\\rightarrow \\ensuremath{\\mathrm{Alg}}_{\\Lambda^{\\mathrm{op}}_{\/[p]}}(\\ensuremath{\\catsingle{C}}^{\\otimes})}$\ninduced by the inclusion $\\tau_p\\colon \\Lambda^\\mathrm{op}_{\/[p]}\\hookrightarrow \\Delta^{\\mathrm{op}}_{\/[p]}$ (see \\cref{example:maps-gen-ns-operad}) admits a fully faithful left adjoint\n\\vspace{-0.2cm}\n\\[\\ensuremath{\\mathrm{Alg}}_{\\Lambda^{\\mathrm{op}}_{\/[p]}}(\\ensuremath{\\catsingle{C}}^{\\otimes})\\xlra{\\tau_{p,!}}  \\ensuremath{\\mathrm{Alg}}_{\\Delta^{\\mathrm{op}}_{\/[p]}}(\\ensuremath{\\catsingle{C}}^{\\otimes})=\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}} )_p\\] by \\cite[Corollary 4.20]{HaugsengMorita}, and $M\\in\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}} )_{[p]}$ is called \\emph{composite} if $M$ is in the essential image of $\\tau_{p,!}$, or equivalently if the counit $\\tau_{p,!}\\tau_p^*M\\rightarrow M$ is an equivalence \\cite[Definition 4.21]{HaugsengMorita}. By \\cite[Corollary 4.38]{HaugsengMorita}, the simplicial structure on the pre-Morita category restricts to a simplicial structure on the full subcategories $\\smash{\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})_{[p]}\\subset \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}} )_{[p]}}$ of composite objects, and by \\cite[Theorem 4.39]{HaugsengMorita}, the result is a double $\\infty$-category---the \\emph{Morita double $\\infty$-category of $\\ensuremath{\\catsingle{C}}$} \\[\n\t\\gls*{alg}\\in\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})\\subseteq\\ensuremath{\\mathrm{Fun}}(\\Delta^\\mathrm{op},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}).\n\\] \nNote that $\\Lambda_{\/[p]}=\\Delta_{\/[p]}$ for $p=0,1$ so $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})_{[p]}\\subseteq \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}})_{[p]}$ is an equality for $p=0,1$, i.e.\\,\n\\begin{equation}\\label{equ:small-simplices-morita}\n\t\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}})_{[0]}=\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})_{[0]}=\\ensuremath{\\mathrm{Ass}}(\\ensuremath{\\catsingle{C}})\\quad\\text{and}\\quad  \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}})_{[1]}=\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})_{[1]}=\\ensuremath{\\mathrm{Bimod}}(\\ensuremath{\\catsingle{C}}).\n\\end{equation}\nIn particular, the mapping $\\infty$-categories between $A,B\\in \\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})_{[0]}$ in the notation of \\cref{sec:mapping-infinity-category} are given as $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})_{A,B}=\\ensuremath{\\mathrm{Bimod}}_{A,B}(\\ensuremath{\\catsingle{C}})$.\n\n\\begin{rem}In \\cite[4.4.3.10, 4.4.3.11]{LurieHA} Lurie describes a Morita double $\\infty$-category $\\ensuremath{\\mathrm{BMod}}(\\ensuremath{\\catsingle{C}})^\\circledast $ for monoidal $\\infty$-categories $\\ensuremath{\\catsingle{C}}$ that admit geometric realisations which are compatible with tensoring with a fixed object on either side (so they in particular admit good relative tensor products). On the level of $0$- and $1$-simplices, $\\ensuremath{\\mathrm{BMod}}(\\ensuremath{\\catsingle{C}})^\\circledast$ and $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})$ agree as a result of \\cref{rem:lurie-bimodules} and it seems likely that there is an equivalence $\\ensuremath{\\mathrm{BMod}}(\\ensuremath{\\catsingle{C}})^\\circledast \\simeq \\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})$ of double $\\infty$-categories, but we do not know a reference for this.\\end{rem}\n\n\n\\subsubsection{Functoriality and monoidality}\\label{sec:morita-functoriality}\nPostcomposition induces a functor\n\\begin{equation}\\label{equ:premorita-functor}\n\t\\overline{\\ensuremath{\\mathrm{ALG}}}(-)\\colon\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})\\longrightarrow \\ensuremath{\\mathrm{Fun}}(\\Delta^\\mathrm{op},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})\n\\end{equation}\nwhich is on $p$-simplices given by $\\ensuremath{\\mathrm{Alg}}_{{\\Delta^\\mathrm{op}}_{\/[p]}}(-)$. The latter preserves limits  \\cite[p.\\,1701]{HaugsengMorita}, so \\eqref{equ:premorita-functor} does as well and in particular induces a functor between commutative monoid objects\n\\[\n\t\\overline{\\ensuremath{\\mathrm{ALG}}}(-)\\colon\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))\\longrightarrow \\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Fun}}(\\Delta^\\mathrm{op},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})).\n\\]\nThe situation for $\\ensuremath{\\mathrm{ALG}}(-)$ is only slightly more complicated. Let $\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})^{\\mathrm{grtp}}\\subset \\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$ the (non-full) subcategory of monoidal categories that admit good relative tensor products and functors that are compatible with them. The latter is made precise in \\cite[Definition 4.18]{HaugsengMorita}, but all we need is that (i) monoidal categories admit good relative tensor products if their underlying categories admit all geometric realisations and these are compatible with tensoring with a fixed object on either side, and that (ii) functors of monoidal categories that preserve geometric realisations are compatible with good relative tensor products. Then $\\ensuremath{\\mathrm{ALG}}(-)$ gives rise to a functor $\\ensuremath{\\mathrm{ALG}}(-)\\colon\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})^{\\mathrm{grtp}}\\longrightarrow \\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$ (see \\cite[Corollary 5.41]{HaugsengMorita}). By \\cite[Lemma 5.38 (iii)]{HaugsengMorita}, the category $\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})^{\\mathrm{grtp}}$ admits products and these are preserved by the forgetful functor $\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})^{\\mathrm{grtp}}\\rightarrow \\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$. Moreover, $\\ensuremath{\\mathrm{ALG}}(-)$ is product-preserving: we may test this on $p$-simplices for $p\\ge0$ and since it has values in double $\\infty$-categories, it suffices to check this for $p=0,1$ where it follows from \\eqref{equ:small-simplices-morita} and the corrresponding fact for $\\overline{\\ensuremath{\\mathrm{ALG}}}(-)$.  $\\ensuremath{\\mathrm{ALG}}(-)$ thus induces a functor on commutative monoid objects:\n\\[\\ensuremath{\\mathrm{ALG}}(-)\\colon\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})^{\\mathrm{grtp}})\\longrightarrow \\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})).\\]\n\n\\subsubsection{Composite algebras in terms of semisimplicial objects}\\label{sec:composite-multisimplicial}\nThe condition on an object $M\\in\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}} )_{[p]}$ to be composite can be reformulated in terms of a certain semisimplicial object which resembles a bar-construction. To explain this, we denote by $\\Delta^{\\mathrm{act}}$ the wide subcategory of $\\Delta$ given by the active maps and by $\\ensuremath{\\catsingle{C}}^{\\otimes,\\mathrm{act}}$ the pullback of $\\ensuremath{\\catsingle{C}}^{\\otimes}\\rightarrow \\Delta^{\\mathrm{op}}$ along the inclusion $\\Delta^{\\mathrm{act},\\mathrm{op}}\\rightarrow \\Delta^{\\mathrm{op}}$. The unique active maps $[1]\\rightarrow [p]$ define a natural transformation from the inclusion $\\Delta^{\\mathrm{act},\\mathrm{op}}\\rightarrow \\Delta^\\mathrm{op}$ to the constant functor at $[1]\\in\\Delta^\\mathrm{op}$, which we can precompose with the projection $\\ensuremath{\\catsingle{C}}^{\\otimes,\\mathrm{act}}\\rightarrow \\Delta^{\\mathrm{act},\\mathrm{op}}$. Taking a cocartesian pushforward (see \\cref{sec:straightening}) of the canonical map $\\ensuremath{\\catsingle{C}}^{\\otimes,\\mathrm{act}}\\rightarrow \\ensuremath{\\catsingle{C}}^{\\otimes}$ along this natural transformation gives a functor\n\\begin{equation}\\label{equ:universal-active-pushforward}\n\t(-)_!\\colon \\ensuremath{\\catsingle{C}}^{\\otimes,\\mathrm{act}}\\longrightarrow \\ensuremath{\\catsingle{C}}_{[1]}^{\\otimes}=\\ensuremath{\\catsingle{C}}.\n\\end{equation}\nNow write $\\Delta^{\\mathrm{act}}_{\/[p]}$ for the wide subcategory of $\\Delta_{\/[p]}$ of those morphisms that map to $\\Delta^{\\mathrm{act}}$ under the projection and $\t\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]}\\subset \\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]}$ for the full subcategory of cellular maps. For $M\\in\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}})_{[p]}\\subset\\ensuremath{\\mathrm{Fun}}_{\\Delta^\\mathrm{op}}(\\Delta^\\mathrm{op}_{\/[p]},\\ensuremath{\\catsingle{C}}^{\\otimes})$ and an object $\\alpha\\colon [q]\\rightarrow[p]$ of $\\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]}$, we consider the composition \n\\begin{equation}\\label{equ:kan-extension-diagram}\n\t((\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]})_{\/\\alpha})^{\\rhd}\\xra{\\mathrm{can}} (\\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]})_{\/\\alpha}\\xra{\\mathrm{pr}}\\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]}\\xra{M}\\ensuremath{\\catsingle{C}}^{\\mathrm{act},\\otimes}\\xra{(-)_!}\\ensuremath{\\catsingle{C}}\n\\end{equation}\nwhere $\\mathrm{pr}$ is the projection, $\\gls*{rhd}$ is the right-cone (this freely adds a terminal object and can be modelled by the join $(-) \\ast \\Delta^0$), and $\\mathrm{can}$ is the extension of the inclusion $\\smash{(\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]})_{\/\\alpha} \\subset (\\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]})_{\/\\alpha}}$ to the cone $((\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]})_{\/\\alpha})^{\\rhd}$ by sending the terminal object to $\\mathrm{id}_\\alpha$.\n\n\\begin{lem}\\label{lem:composite-algebras}For a monoidal $\\infty$-category $\\ensuremath{\\catsingle{C}}$ with good relative tensor products,\n\tan object $M\\in \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}} )_{[p]}$ is composite if and only if  the composition \\eqref{equ:kan-extension-diagram} is a colimit diagram for all $\\alpha$.\n\\end{lem}\n\n\\begin{proof}By definition, $M$ is composite if the counit $\\tau_{p,!}\\tau_p^*M\\rightarrow M$ is an equivalence. Using Proposition 4.16 and Corollary A.60 of \\cite{HaugsengMorita} this is equivalent to asking whether the identity $\\tau_p^*M\\rightarrow \\tau_p^*M$ exhibits $M$ as the operadic left Kan extension of $\\tau_p^*M$ along $\\tau_p$ in the sense of Definition A.56 loc.cit.. By Lemma A.53 loc.cit., this is in turn equivalent to the condition that the functor $((\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]})\\times\\Delta^1)\\sqcup_{(\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]})\\times \\{0\\}}(\\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]})\\times \\{0\\}\\rightarrow \\ensuremath{\\catsingle{C}}$ induced by the restriction of\n\t\\[(\\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]})\\times\\Delta^1\\xlra{\\mathrm{pr}_1}\\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]}\\xra{M}\\ensuremath{\\catsingle{C}}^{\\mathrm{act},\\otimes}\\xra{(-)_!}\\ensuremath{\\catsingle{C}}\\]\n\tto  $\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]}\\times \\Delta^1$ is a left Kan extension in the sense of \\cite[4.3.3.2]{LurieHTT}. As $\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]}\\subset \\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]}$ is a full subcategory inclusion, we can use the simpler characterisation of left Kan extensions from \\cite[4.3.2.2]{LurieHTT}, which is exactly the condition of the statement. \\end{proof}\n\nUnravelling the definitions, the category $(\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]})_{\/\\alpha}$ for $\\alpha \\colon [q] \\to [p]$ is the 1-category whose objects are factorisations of $\\alpha$ into an active map followed by a cellular map,\n\\[\\begin{tikzcd}[column sep=1.5cm]\n\t\\left[q\\right]\\rar{\\text{active}}\\arrow[rr,\"\\alpha\", bend right=20,swap]&\\left[r\\right]\\rar{\\text{cellular}}&\\left[p\\right].\n\\end{tikzcd}\n\\] \nNote that if $\\alpha$ is active, then so must be $[r] \\to [p]$. Given another such factorisation, with middle object $[r']$, a morphism from the factorisation involving $[r]$ to that involving $[r']$ is an active map $[r']\\rightarrow[r]$ that makes everything commute:\n\\[\\begin{tikzcd}[column sep=1.5cm] &  {[r]} \\arrow{rd}{\\text{cellular}} \\arrow{dd}[description]{\\text{active}}& \\\\[-15pt] \n\t[q] \\arrow{ru}{\\text{active}} \\arrow{rd}[swap]{\\text{active}} & & {[p]} \\\\[-15pt]\n\t& {[r']} \\arrow{ru}[swap]{\\text{cellular}} & \\end{tikzcd}\\]\t\nColimits over this category---as appearing in \\eqref{lem:composite-algebras}---can be rephrased in terms of semisimplicial objects that are easier to handle. Making this precise involves the following construction:\n\n\\begin{construction}\\label{const:rhoalpha}Let $\\alpha \\colon [q] \\to [p]$ be an object of $\\Delta^\\mathrm{op}_{\/[p]}$ considered as a sequence $(i_0\\leq\\ldots\\leq i_q) \\subseteq [p]$. Let $J \\subseteq [q-1]$ be the set of indices $j$ for which $i_j < i_{j+1}$ and set $k_{\\alpha} \\coloneqq \\sum_{j \\in J} (i_{j+1}-i_j-1)$. Enumerate the indices in the interval $[i_0,i_q]$ that do \\emph{not} lie in $(i_0,\\ldots,i_q)$ in order as $m_1,\\ldots,m_{k_{\\alpha}}$. Then there is a functor \n\\[\n\t\\rho_\\alpha \\colon (\\Delta^\\mathrm{op})^{k_{\\alpha}} \\longrightarrow (\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]})_{\/\\alpha}\n\\]\ngiven as follows. Writing $\\textstyle{k_\\alpha^{\\vec{a}}\\coloneqq q+\\sum_{i=1}^{k_{\\alpha}}(a_i+1)}$, it sends an object $([a_1],\\ldots,[a_q])\\in(\\Delta^\\mathrm{op})^{k_{\\alpha}}$ to\n\\[\\begin{tikzcd}[column sep=0.8cm]\n\t\\left[q\\right]\\rar{\\alpha_0^{\\vec{a}}}\\arrow[rr,\"\\alpha\", bend right=25,swap]&{[k_\\alpha^{\\vec{a}}]}\\rar{\\alpha_1^{\\vec{a}}}&\\left[p\\right]\n\\end{tikzcd}\\]\nwhere $\\alpha_1^{\\vec{a}}$ is given by the weakly increasing sequence that contains $(i_0\\le \\ldots\\le i_q)$ as well as each $m_j$ repeated $a_j+1$ times. The map $\\alpha_0^{\\vec{a}}$ is the unique injective map such that  $\\alpha^{\\vec{a}}_1\\circ\\alpha^{\\vec{a}}_0=\\alpha$.\\end{construction}\n\n\\begin{ex}\\label{ex:cofinal-alphas}\nIf $\\alpha \\colon [2] \\to [p]$ is given by the sequence $(i\\le i+2\\le i+4)$ then $k_{\\alpha}=2$, the map $\\alpha^{\\vec{a}}_1$ is given by the sequence $(i\\le i+1\\le \\ldots\\le i+1\\le i+2\\le i+3\\le \\ldots\\le i+3\\le i+4)$ where $i+1$ appears $a_1+1$ times and $i+3$ appears $a_2+1$ times, and the map $\\alpha^{\\vec{a}}_0$ is given by the inclusion of $(i\\le i+2\\le i+4)$ into this sequence.\n\\end{ex}\n\nFor later reference, we spell out how $\\rho_\\alpha$ translates under the isomorphism $\\Delta\\cong\\ensuremath{\\cat{Cut}}$.\n\n\\begin{construction}\\label{const:rhoalpha-cut}\nLet $\\alpha \\colon \\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis \\to \\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis$ be an object of $\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\/}$ considered as a sequence $(i_1\\leq\\ldots\\leq i_p)$ with $i_j\\in\\llparenthesis\\hspace{.1em} \\mathring{q}\\hspace{.1em}\\rrparenthesis$. The quantity $k_\\alpha$ is then given by $k_\\alpha\\coloneqq \\#\\{j\\in\\llparenthesis\\hspace{.1em} \\mathring{p{-}1}\\hspace{.1em}\\rrparenthesis\\mid i_j\\in\\llparenthesis\\hspace{.1em} \\mathring{q}\\hspace{.1em}\\rrparenthesis\\text{ and }i_j=i_{j+1}\\}$. We enumerate this set in order by $n_1<\\ldots <n_{k_\\alpha}$. The functor $\\rho_\\alpha$ takes the form\n\\[\n\t\\rho_\\alpha \\colon \\ensuremath{\\cat{Cut}}^{k_{\\alpha}} \\longrightarrow (\\ensuremath{\\cat{Cut}}^{\\mathrm{act},\\mathrm{op}}_{\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\/})_{\/\\alpha}\n\\]\nand can be described as follows: abbreviating $\\textstyle{k_\\alpha^{\\vec{a}}\\coloneqq q+\\sum_{i=1}^{k_{\\alpha}}(a_i+1)}$ as in the $\\Delta^\\mathrm{op}$-case, for an object $\\llparenthesis\\hspace{.1em}\\vec{a}\\hspace{.1em}\\rrparenthesis=(\\llparenthesis\\hspace{.1em} a_1\\hspace{.1em}\\rrparenthesis,\\ldots,\\llparenthesis\\hspace{.1em} a_q\\hspace{.1em}\\rrparenthesis)\\in\\ensuremath{\\cat{Cut}}^{k_{\\alpha}}$, it sends $\\llparenthesis\\hspace{.1em}\\vec{a}\\hspace{.1em}\\rrparenthesis$ to the factorisation\n\\[\\begin{tikzcd}[column sep=0.8cm]\n\t\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\\rar{\\alpha_1^{\\vec{a}}}\\arrow[rr,\"\\alpha\", bend right=25,swap]&\\llparenthesis\\hspace{.1em} k_\\alpha^{\\vec{a}}\\hspace{.1em}\\rrparenthesis \\rar{\\alpha_0^{\\vec{a}}}&\\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis\n\\end{tikzcd}\\]\nwhere $\\alpha_1^{\\vec{a}}\\colon \\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\\rightarrow \\llparenthesis\\hspace{.1em} k_\\alpha^{\\vec{a}}\\hspace{.1em}\\rrparenthesis$ is given by the sequence obtained from the sequence $(i_1\\leq\\ldots\\leq i_p)$ by inserting a gap of length $a_i$ between $i_{n_j}$ and $i_{n_j+1}$ for $j=1,\\ldots, k_\\alpha$, and $\\alpha_0^{\\vec{a}}\\colon \\llparenthesis\\hspace{.1em} k_\\alpha^{\\vec{a}}\\hspace{.1em}\\rrparenthesis\\rightarrow \\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis$ is the unique surjective map such that $\\alpha_0^{\\vec{a}}\\circ \\alpha_1^{\\vec{a}}=\\alpha$.\n\\end{construction}\n\nThe following lemma is a generalisation of \\cite[Lemma 4.17]{HaugsengMorita}.\n\n\\begin{lem}\\label{lem:cofinal-multi-simplicial}The functor $\\rho_\\alpha$ from \\cref{const:rhoalpha} is cofinal.\n\\end{lem}\n\n\\begin{proof}By \\cite[Theorem 4.1.3.1]{LurieHTT} it suffices to prove that $((\\Delta^{\\mathrm{op}})^{k_{\\alpha}})_{X\/}$ has an initial object for all objects $X$ of $(\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]})_{\/\\alpha}$, i.e. for all factorisations $X$\n\\vspace{-0.1cm}\n\\[\n\t[q] \\overset{\\beta} \\longrightarrow [r] \\overset{\\alpha'}\\longrightarrow [p]\n\\]\nof $\\alpha$ into an active $\\beta$ followed by a cellular $\\alpha'$. The category $((\\Delta^\\mathrm{op})^{k_{\\alpha}})_{X\/}$ then has objects given by active maps $\\delta\\colon [k_\\alpha^{\\vec{a}}]\\rightarrow [r]$ such that in \n\\vspace{-0.1cm}\n\\begin{equation}\\label{eqn:factorisation-rho}\n\t[q]\\xra{\\alpha_0^{\\vec{a}}}[k_\\alpha^{\\vec{a}}]\\xra{\\delta} [r] \\xra{\\alpha'} [p]\n\\end{equation}\nthe full composition agrees with $\\alpha$, the composition of the first two arrows with $\\beta$, and the composition of the final two arrows with $\\alpha_1^{\\vec{a}}$. The morphisms are induced by those of $(\\Delta^\\mathrm{op})^{{k_{\\alpha}}}$ via $[k_\\alpha^{\\vec{a}}]$. We now describe an object of this category: define the number $b_j$ by letting $b_j+1$ be the number of times $m_j$ in \\cref{const:rhoalpha} appears in $\\alpha'$ ($b_j\\ge1$ since $\\alpha'$ is cellular and $\\beta$ is active). Then there is a unique map $\\smash{[k_\\alpha^{\\vec{b}}]\\rightarrow [r]}$ that fits in a factorisation as above, and this map is active because $[q] \\to [r]$ is active. For another factorisation \\eqref{eqn:factorisation-rho}, one checks there is unique morphism $([a_1],\\ldots,[a_{k_{\\alpha}}]) \\to ([b_1],\\ldots,[b_{k_{\\alpha}}])$ in $\\Delta^{k_{\\alpha}}$ that induces a morphism of factorisations, so the factorisation we described provides an initial object in $((\\Delta^{\\mathrm{op}})^{{k_{\\alpha}}})_{X\/}$ as wished.\n\\end{proof}\n\n\\begin{ex}In the case of \\cref{ex:cofinal-alphas} and $X$ given by $[2] \\to [6] \\to [p]$ with $[6] \\to [p]$ given by $(i \\leq i \\leq i+1 \\leq i+1 \\leq i+2 \\leq i+3 \\leq i+4)$ (which determines the active morphism $[2] \\to [6]$ uniquely), the initial object in $((\\Delta^{\\mathrm{op}})^{{k_{\\alpha}}})_{X\/}$ is given as follows: we get $k_\\alpha =2$, $([b_1],[b_2]) = ([1],[0])$, $\\smash{k^{\\vec{b}}_\\alpha} = 5$, and $\\delta \\colon [5] \\to [6]$ is $(0 \\leq 2 \\leq 3 \\leq 4 \\leq 5 \\leq 6)$. To see it is initial, note that equivalently it is terminal among pairs $([a_1],[a_2])$ with factorisations\n\t$[2] \\to [k^{\\vec{a}}_\\alpha] \\to [6] \\to [p]$,\n\twhich is true since $\\delta \\colon [5] \\to [6]$ is bijective onto those elements in $[6]$ which do not get mapped to the image of $\\alpha$.\n\\end{ex}\n\nWe now consider the composition \n\\[\n\\begin{tikzcd}[column sep=0.6cm]\n\t(\\Delta^\\mathrm{op}_{\\mathrm{inj}})^{\\rhd}\\rar{\\mathrm{inc}}\\arrow[rrrrr,\"\\eta^\\alpha\",bend right=10,swap]&(\\Delta^\\mathrm{op})^\\rhd\\rar{\\mathrm{diag}}&((\\Delta^\\mathrm{op})^{k(\\alpha)})^\\rhd \\rar{\\rho_\\alpha}&((\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]})_{\/\\alpha})^\\rhd\\rar{\\mathrm{can}}& (\\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]})_{\/\\alpha}\\rar{\\mathrm{pr}}&\\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]}\n\\end{tikzcd}\n\\]\nwhich we abbreviate as $\\eta^\\alpha$; similar for $\\ensuremath{\\cat{Cut}}$ instead of $\\Delta^\\mathrm{op}$. Unravelling the definitions, one checks that $\\eta^{\\alpha}$ maps an object $[a]\\in (\\Delta^\\mathrm{op}_{\\mathrm{inj}})^{k(\\alpha)}$ to $\\alpha_1^{\\vec{a}}\\colon [k_\\alpha^{\\vec{a}}]\\rightarrow [p]$ with $\\vec{a}=(a,\\ldots,a)$ and the cone point to $\\alpha\\colon [q]\\rightarrow [p]$. The unique map from $[a]$ to the cone point is mapped to $\\alpha_0^{\\vec{a}}\\colon [q]\\rightarrow [k_\\alpha^{\\vec{a}}]$. Since the inclusion $\\Delta^\\mathrm{op}_{\\mathrm{inj}}\\subset \\Delta^\\mathrm{op}$ is cofinal \\cite[4.1.1.8]{LurieHTT}, the diagonal $\\Delta^\\mathrm{op}\\rightarrow(\\Delta^\\mathrm{op})^2$ is sifted and hence cofinal \\cite[5.5.8.1, 5.5.8.4]{LurieHTT},  the functor $\\rho_\\alpha$ is cofinal by the previous lemma, and cofinal functors are closed under composition \\cite[4.1.1.3 (2)]{LurieHTT}, the composition $\\Delta^\\mathrm{op}_\\mathrm{inj}\\rightarrow (\\Lambda^{\\mathrm{act},\\mathrm{op}}_{\/[p]})_{\/\\alpha}$ is cofinal. As colimits in $\\infty$-categories are unaffected by precomposition with cofinal functors \\cite[4.1.1.8]{LurieHTT}, we can simplify the condition in \\cref{lem:composite-algebras} further to:\n\n\\begin{cor}\\label{cor:composite-algebras-multisimplicial}For a monoidal $\\infty$-category $\\ensuremath{\\catsingle{C}}$ with good relative tensor products, an object $M\\in \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\catsingle{C}} )_{[p]}$ is composite if and only if for all $\\alpha \\in \\Delta^{\\mathrm{op}}_{\/[p]}$, the following is a colimit diagram\n\t\\[ (\\Delta_\\mathrm{inj}^\\mathrm{op})^\\rhd\\xra{\\eta^\\alpha}\\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]}\\xra{M}\\ensuremath{\\catsingle{C}}^{\\otimes,\\mathrm{act}}\\xra{(-)_!}\\ensuremath{\\catsingle{C}}.\\]\n\\end{cor}\n\n\\subsection{Span and cospan categories}\\label{sec:span-cospan-cats} We summarise the construction of a of a double $\\infty$-category $\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\catsingle{C}})$ of cospans, following \\cite[Section 5]{HaugsengSpans}.\n\n\\subsubsection{Categories of (co)spans}\nFor an $\\infty$-category $\\ensuremath{\\catsingle{C}}$, the \\emph{pre-span simplicial $\\infty$-category of $\\ensuremath{\\catsingle{C}}$} is \\[\\overline{\\ensuremath{\\mathrm{SPAN}}}^+(\\ensuremath{\\catsingle{C}})\\coloneq \\ensuremath{\\mathrm{Fun}}(\\Sigma^\\bullet,\\ensuremath{\\catsingle{C}}) \\in \\ensuremath{\\mathrm{Fun}}(\\Delta^\\mathrm{op},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})\\]\nwhere $\\Sigma^\\bullet \\colon \\Delta \\rightarrow \\ensuremath{\\mathrm{Cat}}$ is defined as follows: on objects it sends $[n] \\in \\Delta$ the poset $\\Sigma^n$ of pairs $(i,j)$ with $0 \\leq i \\leq j \\leq n$, and $(i,j) \\preceq (i',j')$ if and only if $i \\leq i'$ and $j' \\leq j$, and on morphisms sends $\\phi \\colon [n] \\to [m]$ to the functor $\\Sigma^n \\to \\Sigma^m$ given by $(i,j) \\mapsto (\\phi(i),\\phi(j))$. \n\nIf $\\ensuremath{\\catsingle{C}}$ has finite limits then the \\emph{span double $\\infty$-category of $\\ensuremath{\\catsingle{C}}$}\n\\begin{equation}\\label{equ:span-cat}\\ensuremath{\\mathrm{SPAN}}^+(\\ensuremath{\\catsingle{C}}) \\in \\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})\\end{equation}\nis the levelwise full subcategory $\\ensuremath{\\mathrm{SPAN}}^+(\\ensuremath{\\catsingle{C}})\\subset\\overline{\\ensuremath{\\mathrm{SPAN}}}^+(\\ensuremath{\\catsingle{C}})$ of \\emph{cartesian functors}, where $(\\Sigma^p \\to \\ensuremath{\\catsingle{C}})\\in \\overline{\\ensuremath{\\mathrm{SPAN}}}^+(\\ensuremath{\\catsingle{C}})_{[p]}$ is \\emph{cartesian} if the natural map to it from the right Kan extension of its restriction to the full subcategory $\\Lambda^p\\subset \\Sigma^p$ on $(i,j)$ with $j-i \\leq 1$ is an equivalence.  By \\cite[Proposition 5.14]{HaugsengSpans} $\\ensuremath{\\mathrm{SPAN}}^+(\\ensuremath{\\catsingle{C}})$ is indeed a double $\\infty$-category. We have $\\ensuremath{\\mathrm{SPAN}}^+(\\ensuremath{\\catsingle{C}})_{[0]} = \\ensuremath{\\catsingle{C}}$, and arguing as in \\cite[Proposition 8.3]{HaugsengSpans} one sees that the mapping $\\infty$-categories are $\\ensuremath{\\mathrm{SPAN}}^+(\\ensuremath{\\catsingle{C}})_{A,B} \\simeq \\ensuremath{\\catsingle{C}}_{\/A \\times B}$. \n\nDually, one defines the \\emph{cospan double $\\infty$-category} of $\\ensuremath{\\catsingle{C}}$ and its pre-version as\n\\[\\overline{\\ensuremath{\\mathrm{COSPAN}}}^+(\\ensuremath{\\catsingle{C}}^\\mathrm{op}) \\coloneqq \\overline{\\ensuremath{\\mathrm{SPAN}}}^+(\\ensuremath{\\catsingle{C}}^\\mathrm{op})^\\mathrm{op}\\quad\\text{and}\\quad\\gls*{cospan} \\coloneqq \\ensuremath{\\mathrm{SPAN}}^+(\\ensuremath{\\catsingle{C}}^\\mathrm{op})^\\mathrm{op}\\]\nwhere the outer $(-)^\\mathrm{op}$ denotes taking levelwise opposites, and the second definition requires $\\ensuremath{\\catsingle{C}}$ to have finite colimits. The mapping $\\infty$-categories are then given by $\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\catsingle{C}})_{A,B}\\simeq \\ensuremath{\\catsingle{C}}_{A\\sqcup B\/}$.\n\n\\subsubsection{Relation to Morita categories}\\label{sec:alg-cospans}Cospan categories and Morita categories are not unrelated: if $\\ensuremath{\\catsingle{C}}$ has finite colimits, then it has good relative tensor products as in \\cref{sec:composite-algebras} when equipped with the cocartesian symmetric monoidal structure $\\ensuremath{\\catsingle{C}}^{\\sqcup}$ \\cite[Remark 2.5.14]{HaugsengMelaniSafranov}. By Corollaries 2.6.8 and 2.6.10 loc.cit.\\,there is an equivalence of double $\\infty$-categories\n\\begin{equation}\\label{equ:cospan-are-alg} \n\t\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\catsingle{C}})\\simeq \\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}}^{\\sqcup}).\n\\end{equation}\nMoreover, functors that preserve finite colimits are compatible with good relative tensor products, so $\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\catsingle{C}})$ inherits the functoriality and monoidality properties from $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}}^{\\sqcup})$ as discussed in \\cref{sec:morita-functoriality} for monoidal categories with finite colimits and functors that preserve those (there is also an a priori description, but we will not need it). Tracing through the proof one sees that under the equivalence \\eqref{equ:cospan-are-alg}, an object $M \\in \\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})_{[p]}$ is sent to the sequence of cospans\n\\[\\begin{tikzcd}[column sep=.5cm,row sep=0.2cm] \n\t& M(0,1) & &[5pt] \\cdots &[5pt] &[-5pt] M(p{-}1,p) &[-5pt] \\\\[-6pt]\n\tM(0) \\arrow{ru} & & M(1) \\arrow{ru} \\arrow{lu} & & M(p{-}1) \\arrow{ru} \\arrow{lu} & & M(p) \\arrow{lu} \n\\end{tikzcd}\\]\nwhere the map $M(i) \\to M(i,i+1)$ is given by $M(i) \\xra{\\mathrm{inc}} M(i) \\sqcup M(i,i+1) \\xra{\\text{act}} M(i,i+1)$, and similarly for $M(i+1) \\to M(i,i+1)$. Here we used the notation from \\cref{sec:pre-morita}.\n\n\\section{From the bordism to the Morita category}\n\\label{sec:the-functor}\n\nAs part of the introduction, we announced in \\cref{sec:intr-bordism} the construction of a functor\n\\begin{equation}\\label{equ:functor-to-morita-oo-2}\n\t\\gls*{Efunctor} \\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\longrightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)\n\\end{equation}\nof symmetric monoidal $(\\infty,2)$-categories where the domain is an $(\\infty,2)$-category of possibly non-compact $(d-1)$-dimensional manifolds with bordisms as $1$-morphisms and embeddings as $2$-morphisms, and the target is a Morita $(\\infty,2)$-category of the symmetric monoidal $\\infty$-category of presheaves on an $\\infty$-category of disjoint unions of $d$-dimensional open discs with embeddings as morphisms and disjoint union as monoidal structure. What we will actually do is to construct \\eqref{equ:functor-to-morita-oo-2} as a functor between symmetric monoidal double $\\infty$-categories, which is more general by the discussion in \\cref{sec:double-vs-infty2}. \n\n\\medskip\n\n\\begin{center}\\textit{For most of the arguments in the proofs of Theorems~\\ref{bigthm:2-type-invariance}--\\ref{bigthm:nontrivial}, the precise construction of \\eqref{equ:functor-to-morita-oo-2} does not play a role. We summarise the key features in \\cref{sec:functor-e-disc-structure}, so readers who mainly care about Theorems \\ref{bigthm:2-type-invariance}--\\ref{bigthm:nontrivial} may skip this technical section on a first reading.}\n\\end{center}\n\n\\medskip\n\nThe steps we take in this section to construct the functor \\eqref{equ:functor-to-morita-oo-2}  are as follows:\n\n\\begin{enumerate}[leftmargin=1.3cm]\n\t\\item[\\ref{step:bordismcat}] Construct a non-unital double $\\infty$-category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\in\\ensuremath{\\mathrm{Cat}}_{\\mathrm{nu}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$\n\tof possibly non-compact $(d-1)$-manifolds with embeddings and bordisms between them.\n\t\\item[\\ref{step:mand}] Construct a monoidal $\\infty$-category $\\ensuremath{\\icat{M}\\mathrm{an}}_d\\in \\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$ of possibly non-compact $d$-manifolds and embeddings between them, monoidal via disjoint union.\n\t\\item[\\ref{step:functor-to-premorita}]  Construct a morphism \n\t$E^{\\mathrm{geo}} \\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\rightarrow \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)$\n\tof semisimplicial $\\infty$-categories to the pre-Morita category of $\\ensuremath{\\icat{M}\\mathrm{an}}_d$ from \\cref{sec:haugseng-morita}, viewed as a semisimplicial object.\n\t\\item[\\ref{step:composite}]Show that the composition\n\t\\[\n\t\t\\qquad \\quad \\overline{E} \\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}} \\rightarrow \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)\\rightarrow \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d))\\rightarrow\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))\n\t\\]\n\tlands in the Morita category $\\ensuremath{\\icat{M}\\mathrm{od}}(d)\\coloneq \\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))\\subset\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))$. The second map is induced by the Yoneda embedding and the third map by the full subcategory $\\ensuremath{\\icat{D}\\mathrm{isc}}_d\\subset \\ensuremath{\\icat{M}\\mathrm{an}}_d$ on manifolds diffeomorphic to $T\\times\\ensuremath{\\mathbf{R}}^d$ for finite $T$.\n\t\\item[\\ref{step:functor-to-morita}]Argue that $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}$ can be enhanced to a (unital) double $\\infty$-category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\in\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$, and that $\\overline{E}$ can be enhanced to a functor of double $\\infty$-categories as in \\eqref{equ:functor-to-morita-oo-2}.\n\t\\item[\\ref{step:symmetric-monoidal-structure}] Argue that the resulting functor $E \\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d) \\to \\ensuremath{\\icat{M}\\mathrm{od}}(d)$ can be enhanced to a functor of \\emph{symmetric monoidal} double $\\infty$-categories.\n\\end{enumerate}\n\n\\noindent We will conclude the section with some enhancements of the bordism category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$:\n\n\\begin{enumerate}[leftmargin=1.3cm]\n\t\\item[\\ref{step:variants}] Construct variants $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$, $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^\\partial$, and $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d)$ of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ by restricting to compact manifolds, allowing manifolds with boundary, and adding tangential structures.\t\t\n\t\\item[\\ref{step:product}] Construct for a closed $p$-manifold $P$ a map of symmetric monoidal double $\\infty$-categories $P \\times (-) \\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d) \\to \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d+p)$ induced by taking cartesian product with $P$, and extend this construction to the variants from \\ref{step:variants}.\n\\end{enumerate}\n\n\\begin{rem}Some remarks on the construction of the functor \\eqref{equ:functor-to-morita-oo-2}:\n\\begin{enumerate}\n\\item One may ask whether this construction can be ``fully extended''; that is, whether one can upgrade $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ to a symmetric monoidal $(d+1)$-fold $\\infty$-category and the functor $E$ to a map of such objects with target the symmetric monoidal $(d+1)$-fold Morita $\\infty$-category of $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ from \\cite[Section 5]{HaugsengMorita}, which would in particular give a functor of symmetric monoidal $(\\infty,d+1)$-categories. There are no conceptual issues in doing so, but it would involve additional bookkeeping and make our construction less transparent. Since we do not need it to prove the main results, we did not include it.\n\\item We construct \\eqref{equ:functor-to-morita-oo-2} as a functor of symmetric monoidal double $\\infty$-categories, but all later arguments only use the underlying functor of symmetric monoidal $(\\infty,2)$-categories.\n\\item There are at least three constructions of a Morita $(\\infty,2)$-category of a sufficiently nice monoidal $\\infty$-category $\\ensuremath{\\catsingle{C}}$ that for $\\ensuremath{\\catsingle{C}}=\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ might serve as potential targets for \\eqref{equ:functor-to-morita-oo-2}:\n\\begin{enumerate}\n\\item\\label{enum:lurie-model} Lurie's model $\\ensuremath{\\mathrm{BMod}}(\\ensuremath{\\catsingle{C}})$ from \\cite[4.4.3.11]{LurieHA},\n\\item\\label{enum:haugseng-model}  Haugseng's model $\\ensuremath{\\mathrm{ALG}}_1(\\ensuremath{\\catsingle{C}})$ from \\cite[Section 4]{HaugsengMorita}, denoted $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})$ in \\cref{sec:haugseng-morita},\n\\item\\label{enum:Scheimbauer-model}  Scheimbauer's model $\\ensuremath{\\mathrm{Alg}}_1(\\ensuremath{\\catsingle{C}})$ from \\cite[Section 3]{Scheimbauer}.\n\\end{enumerate}\nFor our purposes, Haugseng's version turned out to be the most convenient choice.\n\\item For some of our later arguments, it is crucial that $E$ is defined on the bordism category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ that involves noncompact manifolds; the restriction to the typically considered subcategory $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$ that only involves compact manifolds is not sufficient. If one is mainly interested in a functor from the compact variant $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$ to a Morita category of $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$, then there are other potential routes to a construction, e.g.\\,by modifying a construction of Scheimbauer \\cite{Scheimbauer} or relying on the cobordism hypothesis \\cite{LurieCobordism}.\n\\end{enumerate}\n\\end{rem}\n\nThroughout the following subsections corresponding to the steps above, we generically refer to \\cref{sec:preliminaries} for a recollection of the $\\infty$-categorical concepts and facts involved.\n\n\n\\renewcommand\\thesubsection{\\textbf{Step} $\\circled{\\arabic{subsection}}$}\n\n\\subsection{The bordism category via manifolds with walls}\\label{step:bordismcat}\nWe will construct the non-unital double $\\infty$-category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\in\\ensuremath{\\mathrm{Cat}}_{\\mathrm{nu}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$ as the levelwise coherent nerve of a semisimplicial object in $\\ensuremath{\\cat{Kan}}$-enriched categories \n\\begin{equation}\\label{equ:bordism-cat-simplicial}\n\t\\gls*{ncbordnu} \\in \\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}}_\\mathrm{inj},\\ensuremath{\\cat{sCat}}).\n\\end{equation}\n\n\\begin{convention}\\label{conv:epsilon-conventions}\n\tThroughout this section, we fix a constant $0<\\epsilon<\\tfrac{1}{2}$. We write $\\mathrm{tr}_\\lambda\\colon\\ensuremath{\\mathbf{R}}\\rightarrow\\ensuremath{\\mathbf{R}}$ for the translation by $\\lambda\\in\\ensuremath{\\mathbf{R}}$. For a subset $W\\subset \\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty$, we write \\[W|_A\\coloneqq W\\cap(A\\times\\ensuremath{\\mathbf{R}}^\\infty)\\subset \\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty\\] for subsets $A\\subset\\ensuremath{\\mathbf{R}}$. If $A=\\{a\\}$ is a singleton, we abbreviate $W|_a\\coloneqq W|_{\\{a\\}}$.\n\\end{convention}\n\n\\begin{figure}\n\t\\begin{tikzpicture}[scale=.9]\n\t\\draw [dotted] (-6,0) -- (7,0);\n\t\\node at (6.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\\draw (-5,0) -- (6,0);\n\t\\node at (-3,0) {$\\bullet$};\n\t\\draw [thick,|-|] (-3.5,0) -- (-2.5,0);\n\t\\draw [dashed] (-3,4.5) -- (-3,-1);\n\t\\node at (-3,-.3) [fill=white] {\\tiny $\\mu(0)$};\n\t\\node at (-1,0) {$\\bullet$};\n\t\\draw [thick,|-|] (-1.5,0) -- (-0.5,0);\n\t\\draw [dashed] (-1,4.5) -- (-1,-1);\n\t\\node at (-1,-.3) [fill=white] {\\tiny $\\mu(1)$};\n\t\\node at (2,0) {$\\bullet$};\n\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\\draw [dashed] (2,4.5) -- (2,-1);\n\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(2)$};\n\t\\node at (5,0) {$\\bullet$};\n\t\\draw [thick,|-|] (4.5,0) -- (5.5,0);\n\t\\draw [dashed] (5,4.5) -- (5,-1);\n\t\\node at (5,-.3) [fill=white] {\\tiny $\\mu(3)$};\n\t\n\t\\draw [dotted,Mahogany,thick] (-5.5,3.5) -- (-5,3);\n\t\\draw [dotted,Mahogany,thick] (-5.5,2.5) -- (-5,2);\n\t\\draw [dotted,Mahogany,thick] (-5.5,4.5) -- (-5,4);\n\t\\draw [dotted,Mahogany,thick] (6,1) -- (6.5,1);\n\t\\draw [Mahogany,thick](-5,3) to[out=-45,in=180] (-3.5,2) to[out=0,in=180] (-2.5,2) to[out=0,in=90] (-2,1.5) to[out=-90,in=0] (-2.5,1) to[out=180,in=0] (-3.5,1) to[out=180,in=-45] (-5,2);\n\t\\draw [Mahogany,thick] (-5,4) to[out=-45,in=180] (-3.5,3) to[out=0,in=180] (-.25,3) to[out=0,in=180] (1.5,1) -- (6,1);\n\t\\draw [Mahogany,thick] (3.5,2.5) circle (.75cm);\n\t\\node [Mahogany] at (0,3.5) {$W$};\n\t\\end{tikzpicture}\n\t\\caption{A $[3]$-walled $1$-manifold. The vertical projection is $\\mu$ and the intervals in $\\ensuremath{\\mathbf{R}}$ are the $[\\mu(i)-\\epsilon,\\mu(i)+\\epsilon]$'s, which are disjoint in accordance with \\ref{enum:mfd-wall-i}. The dashed lines indicate the hyperplanes $\\{\\mu(i)\\} \\times \\ensuremath{\\mathbf{R}}^\\infty$. Note that $W$ is transverse to these and a product near them, as imposed in \\ref{enum:mfd-wall-ii} and \\ref{enum:mfd-wall-iii}.}\n\t\\label{fig:mfd-with-walls}\n\\end{figure}\n\nWe first set up some language. A \\emph{$[p]$-walled $d$-manifold} for $[p]\\in\\Delta$ is  a pair $\\gls*{wmu}$ of a $d$-dimensional smooth submanifold $W\\subset\\ensuremath{\\mathbf{R}}\\times \\ensuremath{\\mathbf{R}}^\\infty$ without boundary and an order-preserving map $\\mu\\colon [p]\\rightarrow \\ensuremath{\\mathbf{R}}$ such that the following is satisfied\n\\begin{enumerate}\n\t\\item \\label{enum:mfd-wall-i}$\\mu(i)+\\epsilon<\\mu(i+1)-\\epsilon$ for all $i$,\n\t\\item  \\label{enum:mfd-wall-ii}the projection $\\mathrm{pr}\\colon W\\rightarrow \\ensuremath{\\mathbf{R}}$ to the first coordinate is transverse to $\\mu\\colon [p]\\rightarrow \\ensuremath{\\mathbf{R}}$,\n\t\\item \\label{enum:mfd-wall-iii} $W|_{[\\mu(i)-\\epsilon,\\mu(i)+\\epsilon]}=\\mathrm{tr}_{\\mu(i)}[-\\epsilon,+\\epsilon]\\times W|_{\\mu(i)}$ for all $i$;\n\\end{enumerate}\nsee \\cref{fig:mfd-with-walls} for an example. The space \n\\[\n\t\\ensuremath{\\mathrm{Emb}}\\big((W,\\mu),(W',\\mu')\\big)\\subset \\ensuremath{\\mathrm{Emb}}\\big(W|_{[\\mu(0)-\\epsilon,\\mu(p)+\\epsilon]}, W'|_{[\\mu'(0)-\\epsilon,\\mu'(p)+\\epsilon])}\\big)\n\\] \nof \\emph{embeddings between $[p]$-walled $d$-manifolds} $(W,\\mu)$ and $(W,\\mu')$ is the subspace of those embeddings $\\varphi$ that satisfy the following properties for all $i$:\n\\begin{enumerate}\n\t\\item $\\varphi^{-1}(W|_{[\\mu'(i)+\\epsilon,\\mu'(i+1)-\\epsilon]})=W|_{[\\mu(i)+\\epsilon,\\mu(i+1)-\\epsilon]}$ and $\\varphi^{-1}(W|_{[\\mu'(i)-\\epsilon,\\mu'(i)+\\epsilon]})=W|_{[\\mu(i)-\\epsilon,\\mu(i)+\\epsilon]}$ \n\t\\item the embedding $\\varphi$ restricts on $W|_{[\\mu(i)-\\epsilon,\\mu(i)+\\epsilon]}$ to an embedding of the form \n\t\\[(\\mathrm{tr}_{\\mu'(i)-\\mu(i)}\\times\\varphi_i)\\colon{\\mathrm{tr}_{\\mu(i)}[-\\epsilon,+\\epsilon]}\\times W|_{\\mu(i)}\\lhook\\joinrel\\longrightarrow \\mathrm{tr}_{\\mu'(i)}[-\\epsilon,+\\epsilon]\\times W|_{\\mu'(i)}\n\t\\] for some embedding $\\varphi_i\\in\\ensuremath{\\mathrm{Emb}}(W|_{\\mu(i)}, W|_{\\mu'(i)})$.\n\\end{enumerate} \nUsing this terminology, $\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}$ is defined as the semisimplicial $\\ensuremath{\\cat{Kan}}$-enriched category whose $\\ensuremath{\\cat{Kan}}$-enriched category $\\smash{\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[p]}}$ of $p$-simplices has possibly non-compact $[p]$-walled $d$-manifolds $(W,\\mu)$ as its objects, spaces of embeddings between $[p]$-walled $d$-manifolds as morphisms, and composition given by composition of embeddings. The semisimplicial structure is given by ``forgetting walls'', i.e.\\,by precomposition of $\\mu\\colon [p]\\rightarrow\\ensuremath{\\mathbf{R}}$ with morphisms in $\\Delta_\\mathrm{inj}$.\n\nThe non-unital double $\\infty$-category\n\t\\[\\gls*{ncbordinfnu}\\in\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})\\subset\\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}}_\\mathrm{inj},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})\\]\n\tis now defined as the levelwise coherent nerve of $\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}$, i.e.\\,we have $(\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}})_{[p]}\\coloneqq N_{\\mathrm{coh}}((\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu})_{[p]})$. This implicitly claims that the semisimplicial object $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\in \\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}}_\\mathrm{inj},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$ is indeed a double $\\infty$-category, i.e.\\,that it satisfies the Segal condition.\n\n\\begin{lem}\\label{lem:bord-is-category-object}\n$\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}$ is a non-unital double $\\infty$-category.\n\\end{lem}\n\n\\begin{proof}\nThis is straight-forward, so we will only sketch the proof. One first observes that the Segal maps\t$\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[p]}\\rightarrow \\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[1]}\\times_{\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[0]}}\\ldots \\times_{\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[0]}}\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[1]}$ before taking coherent nerves are Dwyer--Kan equivalences, so weak equivalences in the Bergner model structure from \\cref{sec:scat-vs-qcat} \\ref{enum:bergner-model-structure}. Since the $\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[p]}$ are $\\ensuremath{\\cat{Kan}}$-enriched, they are fibrant in this model structure. Next, one shows that source and target maps $\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[1]}\\rightarrow \\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[0]}$ are Kan fibrations on morphism spaces and isofibrations on homotopy categories, so they are fibrations in the model structure and the pullbacks appearing in the above maps are homotopy pullbacks. Using that the coherent nerve is the right Quillen functor in the Quillen equivalence between the Joyal and the Bergner model structure (see \\cref{sec:scat-vs-qcat} \\ref{enum:bergner-model-structure}) and therefore preserves homotopy pullbacks and weak equivalences between fibrant objects, it follows that the Segal maps $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}_{[p]}\\rightarrow \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}_{[1]}\\times_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}_{[0]}}\\ldots \\times_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}_{[0]}}\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}_{[1]}$ in $\\ensuremath{\\mathrm{Cat}}_\\infty$ are equivalences.\n\\end{proof}\n\n\\subsection{The monoidal category of manifolds and embeddings}\\label{step:mand}\nWe construct the monoidal $\\infty$-category of (possibly noncompact) $d$-manifolds and embeddings between them as a cocartesian fibration $\\gls*{maninf}\\rightarrow \\Delta^{\\mathrm{op}}\\cong \\ensuremath{\\cat{Cut}}$, which we obtain as the coherent nerve of a functor \n\\begin{equation}\\label{equ:man-fibration-simplicial}\n\t\\gls*{mand}\\longrightarrow \\ensuremath{\\cat{Cut}}\n\\end{equation}\nof $\\ensuremath{\\cat{Kan}}$-enriched categories. Objects of $\\ensuremath{\\cat{Man}}_d^{\\sqcup}$ are pairs $(\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis,W)$ of $\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\\in\\ensuremath{\\cat{Cut}}$ and a smooth submanifold $W\\subset \\llparenthesis\\hspace{.1em} \\mathring{p}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty$ without boundary; the distinguished $\\ensuremath{\\mathbf{R}}$-coordinate is not necessary but comes in handy later. To define the space of morphisms, given $A\\subset \\llparenthesis\\hspace{.1em} \\mathring{p}\\hspace{.1em}\\rrparenthesis$, we write \\[W|^A\\coloneqq W\\cap (A\\times\\ensuremath{\\mathbf{R}}\\times \\ensuremath{\\mathbf{R}}^\\infty)\\] to distinguish it from the notation $W|_A$ for $A\\subset \\ensuremath{\\mathbf{R}}$ from \\cref{conv:epsilon-conventions}. Using this, we set\n\\begin{equation}\\label{equ:morphisms-man}\n\t\\textstyle{\\mathrm{Map}_{\\ensuremath{\\cat{Man}}_d^\\sqcup}\\big((\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis,W),(\\llparenthesis\\hspace{.1em} p'\\hspace{.1em}\\rrparenthesis,W')\\big)\\coloneqq\\bigsqcup_ {\\varphi\\in \\mathrm{Map}_\\ensuremath{\\cat{Cut}}(\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} p'\\hspace{.1em}\\rrparenthesis)} \\ensuremath{\\mathrm{Emb}}\\big(W|^{\\varphi^{-1}\\llparenthesis\\hspace{.1em}\\mathring{p}'\\hspace{.1em}\\rrparenthesis}, W')_\\varphi}\n\\end{equation}\nwhere the subscript $(-)_\\varphi$ indicates that we restrict to embeddings that cover $\\varphi$, i.e.\\,that make \n\\begin{center}\n\t\\begin{tikzcd}\n\t\t\\varphi^{-1}\\llparenthesis\\hspace{.1em} \\mathring{p}'\\hspace{.1em}\\rrparenthesis \\times \\ensuremath{\\mathbf{R}} \\times \\ensuremath{\\mathbf{R}}^\\infty \\supset W|^{\\varphi^{-1}\\llparenthesis\\hspace{.1em}\\mathring{p}'\\hspace{.1em}\\rrparenthesis}\\arrow[r,hookrightarrow]\\dar{\\mathrm{pr}_1}&[10pt] W' \\subset \\llparenthesis\\hspace{.1em} \\mathring{p}'\\hspace{.1em}\\rrparenthesis \\times \\ensuremath{\\mathbf{R}} \\times \\ensuremath{\\mathbf{R}}^\\infty \\dar{\\mathrm{pr}_1}\\\\\n\t\t\\varphi^{-1}\\llparenthesis\\hspace{.1em}\\mathring{p}'\\hspace{.1em}\\rrparenthesis\\rar{\\varphi}&\\llparenthesis\\hspace{.1em}\\mathring{p}'\\hspace{.1em}\\rrparenthesis\n\t\\end{tikzcd}\n\\end{center}\ncommute. The composition in $\\ensuremath{\\cat{Man}}_d^\\sqcup$ is induced by the composition in $\\ensuremath{\\cat{Cut}}$ and composition of embeddings. The functor \\eqref{equ:man-fibration-simplicial} sends $(\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis,W)$ to $\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis$. Taking coherent nerves defines $\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup}\\rightarrow \\ensuremath{\\cat{Cut}}\\cong\\Delta^\\mathrm{op}$ which one easily checks to be a monoidal $\\infty$-category using the description in terms of cocartesian fibrations from \\cref{sec:monoidal-cats}.\n\n\\subsubsection{Cocartesian pushforward along active maps}\\label{sec:pushforward-for-man}\nFor later reference, we spell out a model of the cocartesian pushfoward from \\cref{sec:composite-multisimplicial} in the case $\\ensuremath{\\catsingle{C}}^{\\otimes}=\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup}$\n\\[\n(-)_!\\colon \\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup,\\mathrm{act}}\\longrightarrow(\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup})_{[1]}\\eqcolon\\ensuremath{\\icat{M}\\mathrm{an}}_d.\n\\]\nIt is the coherent nerve of a simplicially enriched functor \\begin{equation}\\label{equ:active-pushforward}\\ensuremath{\\cat{Man}}_d^{\\sqcup,\\mathrm{act}}\\rightarrow(\\ensuremath{\\cat{Man}}_d^{\\sqcup})_{[1]}\\eqcolon\\ensuremath{\\cat{Man}}_d,\\end{equation} defined on the pullback of $\\ensuremath{\\cat{Man}}^{\\sqcup}_d$ entlang the inclusion $\\ensuremath{\\cat{Cut}}^{\\mathrm{act}}\\rightarrow\\ensuremath{\\cat{Cut}}$ of the wide subcategory of active maps as in \\cref{section:delta-cut}. The functor \\eqref{equ:active-pushforward} is given by taking ``taking disjoint unions'', using that the restriction to active maps in $\\ensuremath{\\cat{Man}}_d^{\\sqcup,\\mathrm{act}}$ means precisely that the embeddings appearing in \\eqref{equ:morphisms-man} are defined on the whole manifold $W$, not just on a subset depending on the maps $\\varphi$. As a point-set implementation, one can model this ``disjoint unions''-functor as the functor induced by sending submanifolds $W\\subset \\langle \\mathring{p}\\rangle\\times\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty$ to their image under the embedding\n\\[\n\\langle\\mathring{p}\\rangle\\times\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty\\xra{\\mathrm{flip}\\times\\mathrm{id}_{\\ensuremath{\\mathbf{R}}^\\infty}}\\ensuremath{\\mathbf{R}}\\times \\langle\\mathring{p}\\rangle\\times \\ensuremath{\\mathbf{R}}^\\infty\\xra{\\mathrm{id}_\\ensuremath{\\mathbf{R}}\\times\\mathrm{inc}\\times\\mathrm{id}_{\\ensuremath{\\mathbf{R}}^\\infty}}\\ensuremath{\\mathbf{R}}\\times \\ensuremath{\\mathbf{R}}\\times \\ensuremath{\\mathbf{R}}^\\infty\\xra{\\mathrm{id}_\\ensuremath{\\mathbf{R}}\\times\\mathrm{shift}}\\ensuremath{\\mathbf{R}}\\times \\ensuremath{\\mathbf{R}}^\\infty.\n\\]\n\n\\subsection{From $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}$ to the pre-Morita category of manifolds}\\label{step:functor-to-premorita}\nThe construction of the morphism $E^{\\mathrm{geo}} \\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\rightarrow\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)$ goes via the following substeps:\n\\begin{enumerate}[label=(\\alph*)]\n\t\\item \\label{step:functor-to-premorita-language} Set up preparatory language.\n\t\\item \\label{step:functor-to-premorita-thickening} Replace the undercategories $\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/}\\rightarrow \\ensuremath{\\cat{Cut}}$ by a simplicial thickening $\\underline{\\ensuremath{\\cat{Cut}}}_{ \\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/}\\rightarrow \\ensuremath{\\cat{Cut}}$.\n\t\\item \\label{step:functor-to-premorita-simplicial}Construct a functor of semisimplicial objects in $\\ensuremath{\\cat{Kan}}$-enriched categories\n\t\\begin{equation}\\label{equ:functor-to-premorita-simplicial}\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[\\bullet]}\\longrightarrow \\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/ },\\ensuremath{\\cat{Man}}_d^{\\sqcup})\\end{equation}\n\t\\item\\label{step:functor-to-premorita-non-unital} Argue that the resulting functor of semisimplicial $\\infty$-categories\n\t\\begin{equation}\\label{equ:functor-to-premorita-non-unital}\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\longrightarrow \\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/ },\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup})\\end{equation} lands in the levelwise full subcategory $\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)\\subset \\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em}  \\bullet \\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup})$.\n\\end{enumerate}\n\n\\begin{figure}\n\t\\begin{tikzpicture}\n\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\\node at (-4.5,-.3) {\\tiny $L$};\n\t\t\\node at (4,-.3) {\\tiny $R$};\n\t\t\\node at (-.25,-.3) {\\tiny $1 \\in \\llparenthesis\\hspace{.1em} \\mathring{1} \\hspace{.1em}\\rrparenthesis$};\n\t\t\\draw (-5,0) -- (5,0);\n\t\t\\node at (-2.5,0) {$\\bullet$};\n\t\t\\draw [thick,|-|] (-3.5,0) -- (-1.5,0);\n\t\t\\draw [dashed] (-2.5,4.5) -- (-2.5,-1);\n\t\t\\node at (-2.5,-.3) [fill=white] {\\tiny $0 \\in [1]$};\n\t\t\\node at (2,0) {$\\bullet$};\n\t\t\\draw [thick,|-|] (1,0) -- (3,0);\n\t\t\\draw [dashed] (2,4.5) -- (2,-1);\n\t\t\\node at (2,-.3) [fill=white] {\\tiny $1 \\in [1]$};\n\t\t\n\t\t\\draw [dotted,Mahogany] (-5.5,3.5) -- (-5,3);\n\t\t\\draw [dotted,Mahogany] (-5.5,2.5) -- (-5,2);\n\t\t\\draw [dotted,Mahogany] (-5.5,4.5) -- (-5,4);\n\t\t\\draw [dotted,Mahogany] (5,1) -- (5.5,1);\n\t\t\\draw [Mahogany](-5,3) to[out=-45,in=180] (-3.5,2) to[out=0,in=180] (-1.5,2) to[out=0,in=90] (-1,1.5) to[out=-90,in=0] (-1.5,1) to[out=180,in=0] (-3.5,1) to[out=180,in=-45] (-5,2);\n\t\t\\draw [Mahogany] (-5,4) to[out=-45,in=180] (-3.5,3) to[out=0,in=180] (-1.25,3) to[out=0,in=180] (1,1) -- (5,1);\n\t\t\\draw [Mahogany] (4,2.5) circle (.75cm);\n\t\t\n\t\t\\draw [very thick,densely dotted] (-2,2) -- (-1.5,2);\n\t\t\\draw [very thick,densely dotted] (-2,1) -- (-1.5,1);\n\t\t\\draw [very thick,densely dotted] (-2,3) -- (-1.5,3);\n\t\t\\draw [very thick,densely dotted] (1,1) -- (1.5,1);\n\t\t\\draw [very thick,Mahogany] (-1.5,2) to[out=0,in=90] (-1,1.5) to[out=-90,in=0] (-1.5,1);\n\t\t\\draw [very thick,Mahogany] (-1.5,3) to[out=0,in=180] (-1.25,3) to[out=0,in=180] (1,1);\n\t\t\\node [Mahogany] at (-0.35,.9) {$\\mathrm{ch}(W,\\mu)$};\n\t\t\n\t\t\\node [Periwinkle] at (-2.5,1) {$\\blacksquare$};\n\t\t\\node [Periwinkle] at (-2.5,2) {$\\blacksquare$};\n\t\t\\node [Periwinkle] at (-2.5,3) {$\\blacksquare$};\n\t\t\\node [Periwinkle] at (2,1) {$\\blacksquare$};\n\t\t\n\t\t\\node [Periwinkle,fill=white] at (-3.5,4.5) {$\\mathrm{wall}(W,\\mu)$};\n\t\t\\draw [-,white,line width=3pt,shorten >=.4cm,shorten <=.4cm] (-3.5,4.5) -- (-2.5,3);\n\t\t\\draw [->,Periwinkle,shorten >=.4cm,shorten <=.4cm] (-3.5,4.5) -- (-2.5,3);\n\t\t\\draw [-,white,line width=3pt,shorten >=.4cm,shorten <=.4cm] (-3,4.5) to[bend left=30] (2,1);\n\t\t\\draw [->,Periwinkle,shorten >=.4cm,shorten <=.4cm] (-3,4.5) to[bend left=30] (2,1);\n\t\t\n\t\t\\node [fill=white] at (2.5,4.5) {$\\mathrm{coll}(W,\\mu)$};\n\t\t\\draw [-,white,line width=3pt,shorten >=.4cm,shorten <=.4cm] (2.1,4.5) to[bend right=15] (-2,3);\n\t\t\\draw [->,shorten >=.5cm,shorten <=.5cm] (2.1,4.5) to[bend right=15] (-2,3);\n\t\t\\draw [-,white,line width=3pt,shorten >=.4cm,shorten <=.4cm] (2.25,4.4) -- (1.25,1);\n\t\t\\draw [->,shorten >=.3cm,shorten <=.3cm] (2.25,4.4) -- (1.25,1);\n\t\\end{tikzpicture}\n\t\\caption{A $[1]$-walled $1$-manifold. Its walls $\\mathrm{wall}(W,\\mu)$ are the $0$-manifold indicated by the squares, its chambers $\\mathrm{ch}(W,\\mu)$ are the thick region, its collars $\\mathrm{coll}(W,\\mu)$ are the dotted regions, and its thickened chambers $\\mathrm{tch}(W,\\mu)$ are the union of the chambers and the collars.}\n\t\\label{fig:wallsetc}\n\\end{figure}\n\n\\subsubsection*{Substep \\ref{step:functor-to-premorita-language} I: walls and chambers}\nFor a $[p]$-walled $d$-manifold $(W,\\mu)$ as in \\ref{step:bordismcat}, we define \\[\n\t\\gls*{wall} \\subset [p]\\times\\ensuremath{\\mathbf{R}}^\\infty, \\quad \n\t\\gls*{chamber} \\subset \\llparenthesis\\hspace{.1em}\\mathring{p}\\hspace{.1em}\\rrparenthesis \\times\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty,\\quad\\text{and}\\quad\n\t\\gls*{thickchamber}\\subset \\llparenthesis\\hspace{.1em}\\mathring{p}\\hspace{.1em}\\rrparenthesis \\times\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty,\n\\]\n the submanifolds of \\emph{walls}, \\emph{chambers}, and \\emph{thickened chambers} of $(W,\\mu)$, as \\begin{align*}\\mathrm{wall}(W,\\mu)&\\coloneqq \\textstyle{\\bigcup_{i\\in[p]}\\big(\\{i\\}\\times W|_{\\mu(i)}}\\big),\\\\ \\mathrm{ch}(W,\\mu)&\\coloneq\\textstyle{ \\bigcup_{i\\in\\llparenthesis\\hspace{.1em}\\mathring{p}\\hspace{.1em}\\rrparenthesis}\\big(\\{i\\}\\times W|_{[\\mu(i-1)+\\epsilon,\\mu(i)-\\epsilon]}}\\big),\\\\\n\t\\mathrm{tch}(W,\\mu)&\\coloneqq \\textstyle{\\bigcup_{i\\in\\llparenthesis\\hspace{.1em}\\mathring{p}\\hspace{.1em}\\rrparenthesis}\\big(\\{i\\}\\times W|_{(\\mu(i-1)+\\tfrac{\\epsilon}{2},\\mu(i)-\\tfrac{\\epsilon}{2})}}\\big).\n\\end{align*} \nThere is an inclusion $\\mathrm{ch}(W,\\mu)\\subset \\mathrm{tch}(W,\\mu)$ whose complement of the interior we abbreviate as\n\\[\n\t\\gls*{collar}\\coloneqq \\mathrm{tch}(W,\\mu)\\backslash\\mathrm{int}(\\mathrm{ch}(W,\\mu))\\subset \\llparenthesis\\hspace{.1em}\\mathring{p}\\hspace{.1em}\\rrparenthesis \\times\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty.\n\\]\nWe call this the \\emph{collars} of $(W,\\mu)$. Informally, $\\mu$ prescribes hyperplanes $\\{\\mu(i)\\} \\times \\ensuremath{\\mathbf{R}}^\\infty$ intersecting $W$ in the walls, the (thickened) chambers are (thickened) regions between the walls, and the collars are collar neighbourhoods in the thickened chambers; see \\cref{fig:wallsetc} for an example.\n\nGiven in addition a morphism $\\alpha\\in\\mathrm{Map}_\\ensuremath{\\cat{Cut}}(\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis)$, we define the submanifold \n\\[\n\t\\gls*{lab}\\subset \\llparenthesis\\hspace{.1em} \\mathring{q}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty\n\\]\nof \\emph{pieces labelled by $\\alpha$} as the union $\\textstyle{\\mathrm{lab}_\\alpha(W,\\mu)\\coloneqq \\bigcup_{i\\in \\llparenthesis\\hspace{.1em} \\mathring{q}\\hspace{.1em}\\rrparenthesis}\\{i\\}\\times W|_{(\\mu(t_{i-1}^\\alpha)-\\epsilon,\\mu(t_i^\\alpha)+\\epsilon)}}$\nwhere we set $t_i^\\alpha\\coloneqq c^{-1}(\\alpha)(i)$ using the isomorphism \\eqref{equ:cut-iso} and thinking of $\\llparenthesis\\hspace{.1em} \\mathring{q} \\hspace{.1em}\\rrparenthesis=\\{1<\\ldots<q\\}$ as a subset of $[q]=\\{0<\\ldots<q\\}$. Informally, $\\mathrm{lab}_\\alpha(W,\\mu)$ is the set $\\llparenthesis\\hspace{.1em} \\mathring{q}\\hspace{.1em}\\rrparenthesis$ labelled by chambers and thickened walls of $W$ as prescribed by $\\alpha$; see \\cref{fig:lab} for an example.\n\n\\begin{figure}\n\t\\begin{tikzpicture}\n\t\t\t\\begin{scope}[scale=.7,yshift=12cm,xshift=2cm]\n\t\t\t\t\\node at (-2.5,0) {$\\llparenthesis\\hspace{.1em} 3 \\hspace{.1em}\\rrparenthesis$};\n\t\t\t\t\\node at (-2.5,-2) {$\\llparenthesis\\hspace{.1em} 5 \\hspace{.1em}\\rrparenthesis$};\n\t\t\t\t\\draw[->,shorten >=.3cm,shorten <=.3cm] (-2.5,0) -- (-2.5,-2);\n\t\t\t\t\\node at (-2.8,-1) {$\\alpha$};\n\t\t\t\t\n\t\t\t\t\\node [Mahogany] at (0,0) {$\\ast$};\n\t\t\t\t\\node at (0,.4) {\\tiny $L$};\n\t\t\t\t\\node [Mahogany] at (4,0) {$\\ast$};\n\t\t\t\t\\node at (4,.4) {\\tiny $R$};\n\t\t\t\t\\foreach \\i in {1,...,3}\n\t\t\t\t{\n\t\t\t\t\t\\draw[Mahogany,very thick,shorten >=.1cm,shorten <=.1cm] ({\\i-.5},0) -- ({\\i+.5},0);\n\t\t\t\t\t\\node at ({\\i-0},.4) {\\tiny $\\i$};\n\t\t\t\t}\n\t\t\t\t\n\t\t\t\t\\node [Mahogany] at (-1,-2) {$\\ast$};\n\t\t\t\t\\node at (-1,-2.4) {\\tiny $L$};\n\t\t\t\t\\node [Mahogany] at (5,-2) {$\\ast$};\n\t\t\t\t\\node at (5,-2.4) {\\tiny $R$};\n\t\t\t\t\\foreach \\i in {1,...,5}\n\t\t\t\t{\n\t\t\t\t\t\\draw[Mahogany,very thick,shorten >=.1cm,shorten <=.1cm] ({\\i-1.5},-2) -- ({\\i-.5},-2);\n\t\t\t\t\t\\node at ({\\i-1},-2.4) {\\tiny $\\i$};\n\t\t\t\t}\n\t\t\t\t\\draw[Mahogany,->,shorten >=.15cm,shorten <=.15cm] (1,0) -- (1,-2);\n\t\t\t\t\\draw[dotted,shorten >=.2cm,shorten <=.2cm] (0.5,0) -- (0.5,-2);\n\t\t\t\t\\draw[Mahogany,->,shorten >=.15cm,shorten <=.15cm] (2,0) -- (1,-2);\n\t\t\t\t\\draw[dotted,shorten >=.2cm,shorten <=.2cm] (2.5,0) -- (1.5,-2);\n\t\t\t\t\\draw[dotted,shorten >=.2cm,shorten <=.2cm] (2.5,0) -- (2.5,-2);\n\t\t\t\t\\draw[Mahogany,->,shorten >=.15cm,shorten <=.15cm] (3,0) -- (3,-2);\n\t\t\t\t\\draw[dotted,shorten >=.2cm,shorten <=.2cm] (3.5,0) -- (3.5,-2);\n\t\t\t\\end{scope}\n\t\t\t\n\t\t\t\\begin{scope}[scale=.9,yshift=2cm]\n\t\t\t\\begin{scope} \n\t\t\t\t\\clip (-4,-1) rectangle (-2,5);\n\t\t\t\t\n\t\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\t\\node at (-3,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (-3.5,0) -- (-2.5,0);\n\t\t\t\t\\draw [dashed] (-3,3.5) -- (-3,-1);\n\t\t\t\t\\node at (-3,-.3) [fill=white] {\\tiny $\\mu(0)$};\n\t\t\t\t\\node at (-1,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (-1.5,0) -- (-0.5,0);\n\t\t\t\t\\draw [dashed] (-1,3.5) -- (-1,-1);\n\t\t\t\t\\node at (-1,-.3) [fill=white] {\\tiny $\\mu(1)$};\n\t\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(2)$};\n\t\t\t\t\n\t\t\t\t\\draw [(-),Mahogany,thick] (-3.5,3) -- (-2.5,3);\n\t\t\t\t\\draw [(-),Mahogany,thick] (-3.5,2) -- (-2.5,2);\n\t\t\t\t\\draw [(-),Mahogany,thick] (-3.5,1) -- (-2.5,1);\n\t\t\t\t\n\t\t\t\t\\node [fill=white] at (-3,4.2) {$\\mathrm{lab}_\\alpha(W,\\mu)|^{1}$};\n\t\t\t\\end{scope}\n\t\t\t\n\t\t\t\\begin{scope}[xshift=2.5cm]\n\t\t\t\t\\clip (-3.75,-1) rectangle (2.75,5);\n\t\t\t\t\n\t\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\t\\node at (-3,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (-3.5,0) -- (-2.5,0);\n\t\t\t\t\\draw [dashed] (-3,3.5) -- (-3,-1);\n\t\t\t\t\\node at (-3,-.3) [fill=white] {\\tiny $\\mu(0)$};\n\t\t\t\t\\node at (-1,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (-1.5,0) -- (-0.5,0);\n\t\t\t\t\\draw [dashed] (-1,3.5) -- (-1,-1);\n\t\t\t\t\\node at (-1,-.3) [fill=white] {\\tiny $\\mu(1)$};\n\t\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(2)$};\n\t\t\t\t\n\t\t\t\t\\draw [(-),Mahogany,thick] (-3.5,2) -- (-2.5,2) to[out=0,in=90] (-2,1.5) to[out=-90,in=0] (-2.5,1) -- (-3.5,1);\n\t\t\t\t\\draw [(-),Mahogany,thick] (-3.5,3) to[out=0,in=180] (-.25,3) to[out=0,in=180] (1.5,1) -- (2.5,1);\n\t\t\t\t\n\t\t\t\t\\node [fill=white] at (-.5,4.2) {$\\mathrm{lab}_\\alpha(W,\\mu)|^{2}$};\n\t\t\t\\end{scope}\n\t\t\t\n\t\t\t\\begin{scope}[xshift=5cm]\n\t\t\t\t\\clip (1,-1) rectangle (3,5);\n\t\t\t\t\n\t\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\t\\node at (-3,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (-3.5,0) -- (-2.5,0);\n\t\t\t\t\\draw [dashed] (-3,3.5) -- (-3,-1);\n\t\t\t\t\\node at (-3,-.3) [fill=white] {\\tiny $\\mu(0)$};\n\t\t\t\t\\node at (-1,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (-1.5,0) -- (-0.5,0);\n\t\t\t\t\\draw [dashed] (-1,3.5) -- (-1,-1);\n\t\t\t\t\\node at (-1,-.3) [fill=white] {\\tiny $\\mu(1)$};\n\t\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(2)$};\n\t\t\t\t\n\t\t\t\t\\draw [(-),Mahogany,thick] (1.5,1) -- (2.5,1);\n\t\t\t\t\n\t\t\t\t\\node [fill=white] at (2,4.2) {$\\mathrm{lab}_\\alpha(W,\\mu)|^{3}$};\n\t\t\t\\end{scope}\n\t\t\t\n\t\t\t\\begin{scope}[xshift=-3cm,yshift=-6cm]\n\t\t\t\t\\clip (1.25,-1) rectangle (5.75,5);\n\t\t\t\t\n\t\t\t\t\\draw [dotted] (-6,0) -- (7,0);\n\t\t\t\t\\node at (6.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\t\\draw (-5,0) -- (6,0);\n\t\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(2)$};\n\t\t\t\t\\node at (5,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (4.5,0) -- (5.5,0);\n\t\t\t\t\\draw [dashed] (5,3.5) -- (5,-1);\n\t\t\t\t\\node at (5,-.3) [fill=white] {\\tiny $\\mu(3)$};\n\t\t\t\t\n\t\t\t\t\\draw [Mahogany,thick,(-)] (1.5,1) -- (5.5,1);\n\t\t\t\t\\draw [Mahogany,thick] (3.5,2.5) circle (.75cm);\n\t\t\t\t\\node [Mahogany] at (0,3.5) {$W$};\n\t\t\t\t\n\t\t\t\t\\node [fill=white] at (3.5,4.2) {$\\mathrm{lab}_\\alpha(W,\\mu)|^{4}$};\n\t\t\t\\end{scope}\n\t\t\t\n\t\t\t\\begin{scope}[xshift=2.5cm,yshift=-6cm]\n\t\t\t\t\\clip (1,-1) rectangle (3,5);\n\t\t\t\t\n\t\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(3)$};\n\t\t\t\t\n\t\t\t\t\\draw [(-),Mahogany,thick] (1.5,1) -- (2.5,1);\n\t\t\t\t\n\t\t\t\t\\node [fill=white] at (2,4.2) {$\\mathrm{lab}_\\alpha(W,\\mu)|^{5}$};\n\t\t\t\\end{scope}\n\t\t\t\\end{scope}\n\t\\end{tikzpicture}\n\t\\caption{Given the $[3]$-walled $1$-manifold $(W,\\mu)$ of \\cref{fig:mfd-with-walls} and the indicated morphism $\\alpha \\colon \\llparenthesis\\hspace{.1em} 3 \\hspace{.1em}\\rrparenthesis \\to \\llparenthesis\\hspace{.1em} 5 \\hspace{.1em}\\rrparenthesis$, this shows the resulting $\\mathrm{lab}_\\alpha(W,\\mu)$. The following informal description may help: $\\alpha$ tells  which ``parts'' of $\\llparenthesis\\hspace{.1em} 3 \\hspace{.1em}\\rrparenthesis$ to put in which ``box'' of $\\llparenthesis\\hspace{.1em} 5 \\hspace{.1em}\\rrparenthesis$, and if a box is not hit by $\\alpha$ it contains a ``connecting part''.}\n\t\\label{fig:lab}\n\\end{figure}\n\nConstructing the functor \\eqref{equ:functor-to-premorita-simplicial} will require us to describe embeddings out of $\\mathrm{lab}_\\alpha(W,\\mu)$, for which it is helpful to decompose this manifold into two parts as follows: the map $(\\alpha\\times\\mathrm{id}_{\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty})\\colon \\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\\times  \\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty\\rightarrow\\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis\\times  \\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty$ restricts to an embedding \n\\begin{equation}\\label{equ:ch-to-lab}\n\t\\mathrm{tch}(W,\\mu)|^{\\alpha^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}\\hspace{.1em}\\rrparenthesis}\\lhook\\joinrel\\longrightarrow \\mathrm{lab}_\\alpha(W,\\mu)\n\\end{equation}\nusing which we define a submanifold\n\\[\n\t\\gls*{wlab} \\coloneqq \\mathrm{lab}_\\alpha(W,\\mu)\\backslash \\mathrm{int}(\\mathrm{ch}(W)|^{\\alpha^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}\\hspace{.1em}\\rrparenthesis})\\subset \\llparenthesis\\hspace{.1em}\\mathring{q}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty,\n\\]\nof \\emph{thickened walls labelled by $\\alpha$}; see \\cref{fig:wlab} for an example. We have a preferred decomposition\n\\begin{equation}\\label{eq:decompose-lab}\n\t\\mathrm{lab}_\\alpha(W,\\mu)\\cong \\mathrm{ch}(W)|^{\\alpha^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}\\hspace{.1em}\\rrparenthesis} \\cup_\\partial \\mathrm{wlab}_\\alpha(W,\\mu)\n\\end{equation}\nwhere the gluing uses the identification $\\partial(\\mathrm{ch}(W,\\mu)|^{\\alpha^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}\\hspace{.1em}\\rrparenthesis})\\cong\\partial (\\mathrm{wlab}_\\alpha(W,\\mu))$ induced by the restriction of \\eqref{equ:ch-to-lab} to the boundary $\\partial(\\mathrm{ch}(W,\\mu)|^{\\alpha^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}\\hspace{.1em}\\rrparenthesis})$. The restriction \n\\begin{equation}\\label{equ:collar}\n\tc^\\alpha_{(W,\\mu)}\\colon \\mathrm{coll}(W,\\mu)|^{\\alpha^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}\\hspace{.1em}\\rrparenthesis}\\lhook\\joinrel\\longrightarrow \\mathrm{wlab}_\\alpha(W,\\mu)\n\\end{equation}\nof \\eqref{equ:ch-to-lab} to $\\mathrm{coll}(W,\\mu)|^{\\alpha^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}\\hspace{.1em}\\rrparenthesis}$ provides a collar of this boundary. \n\n\\begin{figure}\n\t\\begin{tikzpicture}[scale=.9]\n\t\t\\begin{scope} \n\t\t\t\\clip (-4.3,-1) rectangle (-1.7,5);\n\t\t\t\n\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\\node at (-3,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (-3.5,0) -- (-2.5,0);\n\t\t\t\\draw [dashed] (-3,3.5) -- (-3,-1);\n\t\t\t\\node at (-3,-.3) [fill=white] {\\tiny $\\mu(0)$};\n\t\t\t\\node at (-1,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (-1.5,0) -- (-0.5,0);\n\t\t\t\\draw [dashed] (-1,3.5) -- (-1,-1);\n\t\t\t\\node at (-1,-.3) [fill=white] {\\tiny $\\mu(1)$};\n\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(2)$};\n\t\t\t\n\t\t\t\\draw [(-),Mahogany,thick] (-3.5,3) -- (-2.5,3);\n\t\t\t\\draw [(-),Mahogany,thick] (-3.5,2) -- (-2.5,2);\n\t\t\t\\draw [(-),Mahogany,thick] (-3.5,1) -- (-2.5,1);\n\t\t\t\n\t\t\t\\node [fill=white] at (-3,4.2) {$\\mathrm{wlab}_\\alpha(W,\\mu)|^{1}$};\n\t\t\\end{scope}\n\t\t\n\t\t\\begin{scope}[xshift=2.5cm]\n\t\t\t\\clip (-3.75,-1) rectangle (2.75,5);\n\t\t\t\n\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\\node at (-3,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (-3.5,0) -- (-2.5,0);\n\t\t\t\\draw [dashed] (-3,3.5) -- (-3,-1);\n\t\t\t\\node at (-3,-.3) [fill=white] {\\tiny $\\mu(0)$};\n\t\t\t\\node at (-1,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (-1.5,0) -- (-0.5,0);\n\t\t\t\\draw [dashed] (-1,3.5) -- (-1,-1);\n\t\t\t\\node at (-1,-.3) [fill=white] {\\tiny $\\mu(1)$};\n\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(2)$};\n\t\t\t\n\t\t\t\\draw [(-|,Mahogany,thick] (-3.5,2) -- (-2.5,2);\n\t\t\t\\draw [(-|,Mahogany,thick] (-3.5,1) -- (-2.5,1);\n\t\t\t\\draw [(-|,Mahogany,thick] (-3.5,3) -- (-2.5,3);\n\t\t\t\\draw [|-|,Mahogany,thick] (-1.5,3) -- (-.5,3);\n\t\t\t\\draw [|-),Mahogany,thick] (1.5,1) -- (2.5,1);\n\t\t\t\n\t\t\t\\node [fill=white] at (-.5,4.2) {$\\mathrm{wlab}_\\alpha(W,\\mu)|^{2}$};\n\t\t\\end{scope}\n\t\t\n\t\t\\begin{scope}[xshift=5cm]\n\t\t\t\\clip (.7,-1) rectangle (3.3,5);\n\t\t\t\n\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\\node at (-3,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (-3.5,0) -- (-2.5,0);\n\t\t\t\\draw [dashed] (-3,3.5) -- (-3,-1);\n\t\t\t\\node at (-3,-.3) [fill=white] {\\tiny $\\mu(0)$};\n\t\t\t\\node at (-1,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (-1.5,0) -- (-0.5,0);\n\t\t\t\\draw [dashed] (-1,3.5) -- (-1,-1);\n\t\t\t\\node at (-1,-.3) [fill=white] {\\tiny $\\mu(1)$};\n\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(2)$};\n\t\t\t\n\t\t\t\\draw [(-),Mahogany,thick] (1.5,1) -- (2.5,1);\n\t\t\t\n\t\t\t\\node [fill=white] at (2,4.2) {$\\mathrm{wlab}_\\alpha(W,\\mu)|^{3}$};\n\t\t\\end{scope}\n\t\t\n\t\t\\begin{scope}[xshift=-3cm,yshift=-6cm]\n\t\t\t\\clip (1.25,-1) rectangle (5.75,5);\n\t\t\t\n\t\t\t\\draw [dotted] (-6,0) -- (7,0);\n\t\t\t\\node at (6.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\\draw (-5,0) -- (6,0);\n\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(2)$};\n\t\t\t\\node at (5,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (4.5,0) -- (5.5,0);\n\t\t\t\\draw [dashed] (5,3.5) -- (5,-1);\n\t\t\t\\node at (5,-.3) [fill=white] {\\tiny $\\mu(3)$};\n\t\t\t\n\t\t\t\\draw [Mahogany,thick,(-|] (1.5,1) -- (2.5,1);\n\t\t\t\\draw [Mahogany,thick,|-)] (4.5,1) -- (5.5,1);\n\t\t\t\n\t\t\t\\node [fill=white] at (3.5,4.2) {$\\mathrm{wlab}_\\alpha(W,\\mu)|^{4}$};\n\t\t\\end{scope}\n\t\t\n\t\t\\begin{scope}[xshift=2.5cm,yshift=-6cm]\n\t\t\t\\clip (.7,-1) rectangle (3.3,5);\n\t\t\t\n\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(3)$};\n\t\t\t\n\t\t\t\\draw [(-),Mahogany,thick] (1.5,1) -- (2.5,1);\n\t\t\t\n\t\t\t\\node [fill=white] at (2,4.2) {$\\mathrm{wlab}_\\alpha(W,\\mu)|^{5}$};\n\t\t\\end{scope}\n\t\\end{tikzpicture}\n\t\\caption{The submanifold $\\mathrm{wlab}_\\alpha(W,\\mu)$ for $\\mathrm{lab}_\\alpha(W,\\mu)$ as in \\cref{fig:lab}.}\t\\label{fig:wlab}\\end{figure}\n\n\n\\subsubsection*{Substep \\ref{step:functor-to-premorita-language} II: $\\mathrm{wlab}_\\alpha(-)$ as a pullback}\nUnwrapping the definitions, one sees that $\\mathrm{wlab}_\\alpha(W,\\mu)\\subset \\llparenthesis\\hspace{.1em} \\mathring{q}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty$ is a disjoint union of products of $W|_{\\mu(i)}$ for some $i$ with a (open, half-open, or closed) interval of length $2\\cdot \\epsilon$. More precisely, for $i\\in\\llparenthesis\\hspace{.1em}\\mathring{q}\\hspace{.1em}\\rrparenthesis$ the components $\\mathrm{wlab}_\\alpha(W,\\mu)|^{\\{i\\}}$ of $\\mathrm{wlab}_\\alpha(W,\\mu)$ lying over $i$ are\n$\\mathrm{tr}_{\\mu(t^\\alpha_i)}(-\\epsilon,+\\epsilon)\\times W|_{\\mu(t_i^\\alpha)}$ for $i\\not\\in\\mathrm{im}(\\alpha)$ and \n\\[\n\t\\textstyle{\\Big(\\mathrm{tr}_{\\mu(t^\\alpha_{i-1})}(-\\epsilon,+\\epsilon]\\times W|_{\\mu(t_{i-1}^\\alpha)}\\Big)\\cup \\Big(\\bigcup_{t_{i-1}^\\alpha<j<t_{i}^\\alpha}(\\mathrm{tr}_{\\mu(j)}[-\\epsilon,\\epsilon]\\times W|_{\\mu(j)})\\Big)\\cup \\Big(\\mathrm{tr}_{\\mu(t^\\alpha_i) }[-\\epsilon,+\\epsilon)\\times W|_{\\mu(t_{i}^\\alpha)}\\Big)}\n\\] \nfor $i\\in\\mathrm{im}(\\alpha)$. From this description we see in particular that there are preferred maps \\[\\mathrm{wlab}_\\alpha(W,\\mu)\\rightarrow [p],\\quad \\mathrm{wlab}_\\alpha(W,\\mu)\\rightarrow \\mathrm{wall}(W,\\mu),\\quad\\text{and}\\quad \\mathrm{wlab}_\\alpha(W,\\mu)\\rightarrow \\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}},\\mu)\\]\nwhere we view $(\\ensuremath{\\mathbf{R}},\\mu)$ as a $[p]$-walled $1$-manifold. Indeed, the first map is given by sending the components of $\\mathrm{wlab}_\\alpha(W,\\mu)$ whose first factor is an interval around $\\mu(t_i^{\\alpha})\\in \\ensuremath{\\mathbf{R}}$ to $t_i^{\\alpha}\\in [p]$, the second map is induced by the first map and the projection to $\\ensuremath{\\mathbf{R}}^\\infty$, and the final map is given by the projection to $\\llparenthesis\\hspace{.1em} \\mathring{q}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}$. In particular, this exhibits $\\mathrm{wlab}_\\alpha(W,\\mu)$ as the pullback \n\\begin{equation}\\label{equ:wlab-is-pullback}\n\t\\mathrm{wlab}_\\alpha(W,\\mu)=\\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}},\\mu)\\times_{[p]}\\mathrm{wall}(W,\\mu),\n\\end{equation} \nwhich will be useful to construct embeddings out of $\\mathrm{wlab}_\\alpha(W,\\mu)$; see \\cref{fig:wall-as-pullback} for an example. \n\nIt will also be useful to observe that $\\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}},\\mu)$ is related to $\\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}},\\mu')$ for possibly different $\\mu'\\colon [p]\\rightarrow\\ensuremath{\\mathbf{R}}$ by a preferred diffeomorphism\n\\begin{equation}\\label{equ:wlab-independent-of-walls}\n\t\\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}},\\mu)\\cong \\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}},\\mu'),\n\\end{equation}\nuniquely characterised by requiring it to (i) preserve the order induced by the lexicographical order on $\\llparenthesis\\hspace{.1em} \\mathring{q}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}$ and (ii) agree with translation on each component. For convenience we fix a particular choice of $\\mu$, namely the inclusion $[p]=\\{0,1,\\ldots,p\\}\\subset\\ensuremath{\\mathbf{R}}$ in which case we omit $\\mu$ from the notation, so for instance we abbreviate $\\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}})=\\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}},\\mathrm{inc})$.\n\n\\begin{figure}\n\t\\begin{tikzpicture}[scale=.9]\n\t\t\\begin{scope}[xshift=2.5cm]\n\t\t\t\\clip (-3.75,-1) rectangle (2.75,5);\n\t\t\t\n\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\\node at (-3,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (-3.5,0) -- (-2.5,0);\n\t\t\t\\draw [dashed] (-3,3.5) -- (-3,-1);\n\t\t\t\\node at (-3,-.3) [fill=white] {\\tiny $\\mu(0)$};\n\t\t\t\\node at (-1,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (-1.5,0) -- (-0.5,0);\n\t\t\t\\draw [dashed] (-1,3.5) -- (-1,-1);\n\t\t\t\\node at (-1,-.3) [fill=white] {\\tiny $\\mu(1)$};\n\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(2)$};\n\t\t\t\n\t\t\t\\draw [(-|,Mahogany,thick] (-3.5,2) -- (-2.5,2);\n\t\t\t\\draw [(-|,Mahogany,thick] (-3.5,1) -- (-2.5,1);\n\t\t\t\\draw [(-|,Mahogany,thick] (-3.5,3) -- (-2.5,3);\n\t\t\t\\draw [|-|,Mahogany,thick] (-1.5,3) -- (-.5,3);\n\t\t\t\\draw [|-),Mahogany,thick] (1.5,1) -- (2.5,1);\n\t\t\t\n\t\t\t\\node [fill=white] at (-.5,4.2) {$\\mathrm{wlab}_\\alpha(W,\\mu)|^{2}$};\n\t\t\\end{scope}\n\t\t\n\t\t\\draw [->] (2,-1.3) -- (2,-2);\n\t\t\\draw [->] (8.5,-1.3) -- (8.5,-2);\n\t\t\\draw [->] (5.4,2) -- (6.1,2);\n\t\t\\draw [->] (5.4,-3) -- (6.1,-3);\n\t\t\n\t\t\\begin{scope}[xshift=2.5cm,yshift=-3.5cm]\n\t\t\t\\clip (-3.75,-1) rectangle (2.75,2);\n\t\t\t\n\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\n\t\t\t\\draw [(-|,Mahogany,thick] (-3.5,0) -- (-2.5,0);\n\t\t\t\\draw [|-|,Mahogany,thick] (-1.5,0) -- (-.5,0);\n\t\t\t\\draw [|-),Mahogany,thick] (1.5,0) -- (2.5,0);\n\t\t\t\n\t\t\t\\node [fill=white] at (-.5,.8) {$\\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}},\\mu)|^{2}$};\n\t\t\\end{scope}\n\t\t\n\t\t\\begin{scope}[xshift=9cm,yshift=-3.5cm,xscale=.6]\n\t\t\t\\clip (-3.75,-1) rectangle (2.75,2);\n\t\t\t\n\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\n\t\t\t\\node at (-3,0) [fill=white] {$0$};\n\t\t\t\\node at (-1,0) [fill=white] {$1$};\n\t\t\t\\node at (2,0) [fill=white] {$2$};\n\t\t\t\n\t\t\t\\node [fill=white] at (-.5,.8) {$[3]$};\n\t\t\\end{scope}\n\t\t\n\t\t\\begin{scope}[xshift=9cm,xscale=.6]\n\t\t\t\\clip (-3.75,-1) rectangle (2.75,5);\n\t\t\t\n\t\t\t\\draw [dotted] (-6,0) -- (6,0);\n\t\t\t\\node at (5.5,-.3) {\\tiny $\\ensuremath{\\mathbf{R}}$};\n\t\t\t\\draw (-5,0) -- (5,0);\n\t\t\t\\node at (-3,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (-3.5,0) -- (-2.5,0);\n\t\t\t\\draw [dashed] (-3,3.5) -- (-3,-1);\n\t\t\t\\node at (-3,-.3) [fill=white] {\\tiny $\\mu(0)$};\n\t\t\t\\node at (-1,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (-1.5,0) -- (-0.5,0);\n\t\t\t\\draw [dashed] (-1,3.5) -- (-1,-1);\n\t\t\t\\node at (-1,-.3) [fill=white] {\\tiny $\\mu(1)$};\n\t\t\t\\node at (2,0) {$\\bullet$};\n\t\t\t\\draw [thick,|-|] (1.5,0) -- (2.5,0);\n\t\t\t\\draw [dashed] (2,3.5) -- (2,-1);\n\t\t\t\\node at (2,-.3) [fill=white] {\\tiny $\\mu(2)$};\n\t\t\t\n\t\t\t\\node at (-3,2) [Periwinkle] {$\\blacksquare$};\n\t\t\t\\node at (-3,1) [Periwinkle] {$\\blacksquare$};\n\t\t\t\\node at (-3,3) [Periwinkle]{$\\blacksquare$};\n\t\t\t\\node at (-1,3) [Periwinkle]{$\\blacksquare$};\n\t\t\t\\node at (2,1) [Periwinkle]{$\\blacksquare$};\n\t\t\t\n\t\t\t\\node [fill=white] at (-.5,4.2) {$\\mathrm{wall}(W,\\mu)$};\n\t\t\\end{scope}\n\t\\end{tikzpicture}\n\t\\caption{The pullback decomposition \\eqref{equ:wlab-is-pullback} for one of the part of $\\mathrm{wlab}_\\alpha(W,\\mu)$ from \\cref{fig:wlab}. Note that $\\mathrm{wall}(W,\\mu)$ and $[3]$ are larger than pictured; we have only included the parts that are relevant for this pullback.}\n\t\\label{fig:wall-as-pullback}\n\\end{figure}\n\n\\subsubsection*{Substep \\ref{step:functor-to-premorita-thickening}: Thickening}\nAs a next step, we replace the undercategory functor\n\\begin{equation}\\label{overcat}\n\t\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/}\\colon \\ensuremath{\\cat{Cut}}^{\\mathrm{op}}\\longrightarrow \\ensuremath{\\cat{sCat}}_{\/ \\ensuremath{\\cat{Cut}}}\n\\end{equation}\nby a simplicial thickening after precomposition with the inclusion $\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^{\\mathrm{op}}\\rightarrow \\ensuremath{\\cat{Cut}}^{\\mathrm{op}}$ of the (opposite of) the wide subcategory of surjective morphisms. By ``simplicial thickening'', we mean a functor whose values need no longer be discrete categories and which comes with a natural transformation to $\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/}$ that is a levelwise Dwyer--Kan equivalence. \n\nWe first define a $\\ensuremath{\\cat{Kan}}$-enriched category $\\gls*{cutkan}$ that is Dwyer--Kan equivalent to $\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}$. Its objects are the same as those of $\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}$, that is, morphisms $\\alpha \\colon \\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis \\to \\llparenthesis\\hspace{.1em} q \\hspace{.1em}\\rrparenthesis$ in $\\ensuremath{\\cat{Cut}}$. The space of morphisms from the object $\\alpha\\colon \\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\\rightarrow \\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis$ to the object $\\alpha'\\colon \\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\\rightarrow \\llparenthesis\\hspace{.1em} q'\\hspace{.1em}\\rrparenthesis$ is \n\\[\n\t\\textstyle{\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}}(\\alpha,\\alpha')\\coloneqq \\bigsqcup_{\\gamma\\in \\mathrm{Map}_{\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}}(\\alpha,\\alpha')}  \\ensuremath{\\mathrm{Emb}}\\big(\\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}})|^{{\\gamma^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q'}\\hspace{.1em}\\rrparenthesis}},\\mathrm{wlab}_{\\alpha'}(\\ensuremath{\\mathbf{R}})\\big)_\\gamma}\n\\]\nwhere the subscript $\\gamma$ indicates that we restrict to embeddings $\\overline{\\gamma}$ that \n\\begin{enumerate}\n\t\\item make the diagrams \n\t\\[\\qquad \\begin{tikzcd}[column sep=0.4cm,]\n\t\t\\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}})|^{\\gamma^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q'}\\hspace{.1em}\\rrparenthesis}\\dar\\arrow[r,hook,\"\\overline{\\gamma}\"]& \\mathrm{wlab}_{\\alpha'}(\\ensuremath{\\mathbf{R}})\\dar\\\\\n\t\t\\gamma^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q'}\\hspace{.1em}\\rrparenthesis\\rar{\\gamma}&\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis\n\t\\end{tikzcd}\\quad \\begin{tikzcd}[column sep=-0.4cm]\n\t\t\\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}})|^{\\gamma^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q'}\\hspace{.1em}\\rrparenthesis}\\arrow[rr,\"\\overline{\\gamma}\",hook]&& \\mathrm{wlab}_{\\alpha'}(\\ensuremath{\\mathbf{R}})\\\\\n\t\t&\\mathrm{coll}(\\ensuremath{\\mathbf{R}})|^{\\alpha'^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}\\hspace{.1em}\\rrparenthesis}\\arrow[ur,\"c^{\\alpha'}_{\\ensuremath{\\mathbf{R}}}\",swap,hook']\\arrow[ul,\"c^\\alpha_{\\ensuremath{\\mathbf{R}}}\",hook]&\n\t\\end{tikzcd}\n\t\\]\n\tcommute, i.e.\\,they cover $\\gamma$ and preserve the collars \\eqref{equ:collar}, and\n\t\\item are order-preserving with respect to lexicographical order on $\\llparenthesis\\hspace{.1em} \\mathring{q}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}$ and $\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}$.\n\\end{enumerate}\nThe composition in $\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}$ is induced by the composition in $\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}$, forgetting components, and composition of embeddings. By construction, there is a forgetful functor $\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}\\longrightarrow\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}$ which is a Dwyer--Kan equivalence as a result of the contractibility of the space of monotonous embeddings between connected intervals. Postcomposing this functor with the projection $\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}\\rightarrow \\ensuremath{\\cat{Cut}}$ and varying $p$, we obtain a functor \\[\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/} \\colon \\ensuremath{\\cat{Cut}}_\\mathrm{sur}^{\\mathrm{op}}\\longrightarrow \\ensuremath{\\cat{sCat}}_{\/ \\ensuremath{\\cat{Cut}}}\\] with a natural transformation to \\eqref{overcat} that consists of the Dwyer--Kan equivalences just discussed.\n\n\n\\subsubsection*{Substep \\ref{step:functor-to-premorita-simplicial}: $E^{\\mathrm{geo}}$ on the level of $\\ensuremath{\\cat{Kan}}$-enriched categories}\n\nWe now turn towards the construction of a functor of semisimplicial $\\ensuremath{\\cat{Kan}}$-enriched categories\n\\begin{equation}\\label{equ:psi-simplicially}\n\tE^{\\mathrm{geo}}_{[\\bullet]}\\colon \\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[\\bullet]}\\longrightarrow\\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}\\big(\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\cat{Man}}_d^{\\sqcup}\\big).\n\\end{equation}\nThe value of $E^{\\mathrm{geo}}_{[p]}$ at $(W,\\mu)\\in(\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu})_{[p]}$ is the functor\n\\begin{equation}\\label{equ:psi-single-manifold}\n\tE^{\\mathrm{geo}}_{[p]}(W,\\mu)\\colon \\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}\\longrightarrow \\ensuremath{\\cat{Man}}_d^{\\sqcup}\n\\end{equation}\nover $\\ensuremath{\\cat{Cut}}$ defined as follows: on objects, it maps $(\\alpha\\colon \\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis \\to \\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis)$ to $(\\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis,\\mathrm{lab}_\\alpha(W,\\mu))$. On a morphism given by a pair $(\\gamma,\\overline{\\gamma})$ of a morphism $\\gamma \\colon \\llparenthesis\\hspace{.1em} q \\hspace{.1em}\\rrparenthesis \\to \\llparenthesis\\hspace{.1em} q' \\hspace{.1em}\\rrparenthesis$ under $\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis$ in $\\ensuremath{\\cat{Cut}}$ and an embedding $\\overline{\\gamma}\\in\\ensuremath{\\mathrm{Emb}}(\\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}})|^{{\\gamma^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q'}\\hspace{.1em}\\rrparenthesis}},\\mathrm{wlab}_{\\alpha'}(\\ensuremath{\\mathbf{R}}))_\\gamma$, it is given by the embedding\n\\[\n\tE^{\\mathrm{geo}}_{[p]}(W,\\mu)(\\overline{\\gamma})\\colon\\mathrm{lab}_\\alpha(W,\\mu)|^{{\\gamma^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q'}\\hspace{.1em}\\rrparenthesis}}\\lhook\\joinrel\\longrightarrow \\mathrm{lab}_{\\alpha'}(W,\\mu)\n\\] \nover $\\gamma$ constructed via the following recipe: using the decomposition \\eqref{eq:decompose-lab} and $\\alpha^{-1}(\\gamma^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q'}\\hspace{.1em}\\rrparenthesis) = \\alpha'^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q'}\\hspace{.1em}\\rrparenthesis$, the embedding $E^{\\mathrm{geo}}_{[p]}(W,\\mu)(\\overline{\\gamma})$ is of the form\n\\[\n\t\\mathrm{ch}(W,\\mu)|^{\\alpha'^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q'}\\hspace{.1em}\\rrparenthesis} \\cup_\\partial \\mathrm{wlab}_\\alpha(W,\\mu)|^{\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis}\\lhook\\joinrel\\longrightarrow \\mathrm{ch}(W,\\mu)|^{\\alpha'^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q'}\\hspace{.1em}\\rrparenthesis} \\cup_\\partial \\mathrm{wlab}_{\\alpha'}(W,\\mu).\n\\]\nOn $\\mathrm{ch}(W,\\mu)|^{\\alpha'^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q'}\\hspace{.1em}\\rrparenthesis}$ we declare it to be the identity and on the complement we use the pullback description \\eqref{equ:wlab-is-pullback} and the translations \\eqref{equ:wlab-independent-of-walls} to define it via the commutative diagram\n\\[\\begin{tikzcd}[row sep=0.2cm,column sep=2cm,ar symbol\/.style = {draw=none,\"\\textstyle#1\" description,sloped},\n\tiso\/.style = {ar symbol={\\cong}}]\n\t\\mathrm{wlab}_\\alpha(W,\\mu)|^{\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis}\\arrow[r,hookrightarrow,\"{E^{\\mathrm{geo}}_{[p]}(W,\\mu)(\\overline{\\gamma})}\"]\\arrow[d,equal]&\\mathrm{wlab}_{\\alpha'}(W,\\mu)\\arrow[d,equal]\\\\\n\t\\mathrm{wlab}_{\\alpha}(\\ensuremath{\\mathbf{R}},\\mu)|^{\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis}\\times_{[p]}\\mathrm{wall}(W,\\mu)\\arrow[d,iso]&\\mathrm{wlab}_{\\alpha'}(\\ensuremath{\\mathbf{R}},\\mu)\\times_{[p]}\\mathrm{wall}(W,\\mu)\\arrow[d,iso]\\\\\n\t\\mathrm{wlab}_{\\alpha}(\\ensuremath{\\mathbf{R}})|^{\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis}\\times_{[p]}\\mathrm{wall}(W,\\mu)\\arrow[r,hookrightarrow,\"{(\\overline{\\gamma},\\mathrm{id})}\"]&\\mathrm{wlab}_{\\alpha'}(\\ensuremath{\\mathbf{R}})\\times_{[p]}\\mathrm{wall}(W,\\mu).\n\\end{tikzcd}\\]\nThis finishes the construction of the functor $E^{\\mathrm{geo}}_{[p]}(W,\\mu)\\colon \\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}\\rightarrow \\ensuremath{\\cat{Man}}_d^{\\sqcup}$. Note that it commutes with the functors to $\\ensuremath{\\cat{Cut}}$ by construction. \n\nHaving defined $E^{\\mathrm{geo}}_{[p]}$ on objects, defining it on morphisms amounts to specifying maps\n\\[\n\t\\begin{tikzcd}[row sep=0.4cm, column sep=-0.2cm] \\ensuremath{\\mathrm{Emb}}\\big((W,\\mu),(W',\\mu')\\big) \\dar & \\\\\\ensuremath{\\cat{Nat}}_{\\ensuremath{\\cat{Cut}}}(E^{\\mathrm{geo}}_{[p]}(W,\\mu),E^{\\mathrm{geo}}_{[p]}(W',\\mu'))&\\subset \t\\bigsqcup_{q,\\alpha\\in\\mathrm{Map}_{\\ensuremath{\\cat{Cut}}}(\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis)}\n\t\\ensuremath{\\mathrm{Emb}}\\big(\\mathrm{lab}_\\alpha(W,\\mu),\\mathrm{lab}_\\alpha(W',\\mu')\\big)\\end{tikzcd}\n\\]\nwhere $\\ensuremath{\\cat{Nat}}_\\ensuremath{\\cat{Cut}}(-,-)$ is the hom-functor in the $\\ensuremath{\\cat{Kan}}$-enriched category $\\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}\\big(\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\cat{Man}}_d^{\\sqcup}\\big)$, i.e.\\,the space of natural transformations covering the identity on $\\ensuremath{\\cat{Cut}}$. These maps are induced by the evident naturality of the $\\mathrm{lab}_\\alpha(-)$-construction in embeddings of $[p]$-walled $d$-manifolds. \n\nTo finish the construction of \\eqref{equ:psi-simplicially}, we have to argue that the $E^{\\mathrm{geo}}_{[p]}$'s assemble to a morphism of semisimplicial objects in $\\ensuremath{\\cat{Kan}}$-enriched categories as in \\eqref{equ:psi-simplicially}. But this is merely a case of going through the definitions; ultimately it amounts to the identity $\\mathrm{lab}_{\\beta\\circ c(\\delta)}(W,\\mu)=\\mathrm{lab}_\\beta(W,\\mu\\circ\\delta)$.\n\n\\subsubsection*{Substep \\ref{step:functor-to-premorita-non-unital}: $E^{\\mathrm{geo}}$ on the level of $\\infty$-categories}\nTaking coherent nerves, we obtain \n\\[\\gls*{cutinf} \\coloneqq N_{\\mathrm{coh}}(\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/}) \\in \\ensuremath{\\mathrm{Fun}}(\\Delta^\\mathrm{op}_\\mathrm{inj},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}),\\]\nwhich comes with an equivalence to $\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/}\\cong \\Delta_{\/[\\bullet]}^\\mathrm{op}$ induced by the equivalence $\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/}\\simeq \\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/}$ from Substep \\ref{step:functor-to-premorita-thickening}. From $E^{\\mathrm{geo}} _{[\\bullet]}$ we obtain a morphism of semisimplicial objects in $\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$ \n\\begin{equation}\\label{equ:bord-to-pre-morita-nonalg}\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\longrightarrow \\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\underline{\\ensuremath{\\icat{C}\\mathrm{ut}}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup})\\simeq \\ensuremath{\\mathrm{Fun}}_{\\Delta^\\mathrm{op}}(\\Delta_{\/[\\bullet]}^\\mathrm{op},\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup})\n\\end{equation}\ngiven by postcomposing the coherent nerve applied to \\eqref{equ:psi-simplicially} with the canonical map\n\\[N_{\\mathrm{coh}}\\big(\\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\cat{Man}}_d^{\\sqcup})\\big)\\longrightarrow \\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\underline{\\ensuremath{\\icat{C}\\mathrm{ut}}}_{\\llparenthesis\\hspace{.1em} \\bullet \\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup})\\simeq \\ensuremath{\\mathrm{Fun}}_{\\Delta^\\mathrm{op}}(\\Delta_{\/[\\bullet]}^\\mathrm{op},\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup}),\\]\nfrom Property \\ref{enum:coherent-functor} of \\cref{sec:coherent-nerve-props}.\n\n\\begin{lem}\\label{lem:image-in-premorita}The image of the functor \\eqref{equ:bord-to-pre-morita-nonalg} lies in the levelwise full subcategory $\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d) \\subset \\ensuremath{\\mathrm{Fun}}_{\\Delta^\\mathrm{op}}(\\Delta_{\/[\\bullet]}^\\mathrm{op},\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup})$ from \\cref{sec:pre-morita}.\n\\end{lem}\n\n\\begin{proof}\nIn view of \\cref{fact:cocart-from-s-cat}, it suffices to show that for a $[p]$-walled manifold $(W,\\mu)$, and objects $\\alpha\\colon \\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis \\rightarrow \\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis$ and $\\alpha'\\colon \\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis \\rightarrow \\llparenthesis\\hspace{.1em} q'\\hspace{.1em}\\rrparenthesis$, the functor\n$\n\tE^{\\mathrm{geo}}_{[p]}(W,\\mu)\\colon\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}\\rightarrow \\ensuremath{\\cat{Man}}_d^{\\sqcup}\n$  of $\\ensuremath{\\cat{Kan}}$-enriched categories sends embeddings $\\overline{\\gamma}\\in\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}}(\\alpha,\\alpha')$ whose underlying map $\\gamma\\colon \\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis \\rightarrow \\llparenthesis\\hspace{.1em} q'\\hspace{.1em}\\rrparenthesis$ is inert to cocartesian morphisms in $\\ensuremath{\\cat{Man}}_d^{\\sqcup}$ with respect to  the projection $\\ensuremath{\\cat{Man}}_d^{\\sqcup}\\rightarrow \\ensuremath{\\cat{Cut}}$. In other words, for objects $( Z,\\llparenthesis\\hspace{.1em} q''\\hspace{.1em}\\rrparenthesis)\\in\\ensuremath{\\cat{Man}}_d^{\\sqcup}$, we need to check that the square of Kan-complexes\n\\[\n\\hspace{-0.3cm}\n\\begin{tikzcd}[column sep=0.6cm]\n\t\\mathrm{Map}_{\\ensuremath{\\cat{Man}}_d^{\\sqcup}}\\Big((\\llparenthesis\\hspace{.1em} q'\\hspace{.1em}\\rrparenthesis,\\mathrm{lab}_{\\alpha'}(W,\\mu)),(\\llparenthesis\\hspace{.1em} q''\\hspace{.1em}\\rrparenthesis,Z)\\Big)\\rar{E^{\\mathrm{geo}}_{[p]}(W,\\mu)(\\overline{\\gamma})^*}\\arrow[d,equal]&\\mathrm{Map}_{\\ensuremath{\\cat{Man}}_d^{\\sqcup}}\\Big((\\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis,\\mathrm{lab}_{\\alpha}(W,\\mu)), (\\llparenthesis\\hspace{.1em} q''\\hspace{.1em}\\rrparenthesis,Z)\\Big)\\arrow[d,equal]\\\\[-0.45cm]\n\t\\underset{\\varphi\\in \\mathrm{Map}_\\ensuremath{\\cat{Cut}}(\\llparenthesis\\hspace{.1em} q'\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} q''\\hspace{.1em}\\rrparenthesis)}\\bigsqcup \\ensuremath{\\mathrm{Emb}}\\big(\\mathrm{lab}_{\\alpha'}(W,\\mu)|^{\\varphi^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}''\\hspace{.1em}\\rrparenthesis}, Z)_\\varphi\\dar& \\underset{\\psi\\in \\mathrm{Map}_\\ensuremath{\\cat{Cut}}(\\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} q''\\hspace{.1em}\\rrparenthesis)}\\bigsqcup \\ensuremath{\\mathrm{Emb}}\\big(\\mathrm{lab}_{\\alpha}(W,\\mu)|^{\\psi^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}''\\hspace{.1em}\\rrparenthesis}, Z)_\\psi\\dar\\\\\n\t\\mathrm{Map}_\\ensuremath{\\cat{Cut}}(\\llparenthesis\\hspace{.1em} q'\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} q''\\hspace{.1em}\\rrparenthesis)\\rar{\\gamma^*}& \\mathrm{Map}_\\ensuremath{\\cat{Cut}}(\\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} q''\\hspace{.1em}\\rrparenthesis)\n\t\\end{tikzcd}\n\\]\nis homotopy cartesian. Taking vertical homotopy fibres, it suffices to show that the maps\n\\[\n\tE^{\\mathrm{geo}}_p(W,\\mu)(\\overline{\\gamma})^*\\colon \\ensuremath{\\mathrm{Emb}}\\big(\\mathrm{lab}_{\\alpha'}(W,\\mu)|^{\\varphi^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}''\\hspace{.1em}\\rrparenthesis}, Z)_\\varphi\\longrightarrow\\ensuremath{\\mathrm{Emb}}\\big(\\mathrm{lab}_{\\alpha}(W,\\mu)|^{(\\varphi\\circ\\gamma)^{-1}\\llparenthesis\\hspace{.1em}\\mathring{q}''\\hspace{.1em}\\rrparenthesis}, Z)_{\\varphi\\circ\\gamma}\n\\]\nare weak equivalences. Since $\\gamma$ is inert, the restricted map $\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q}'\\hspace{.1em}\\rrparenthesis\\rightarrow \\llparenthesis\\hspace{.1em} \\mathring{q}'\\hspace{.1em}\\rrparenthesis$ is bijective, so it suffices to show that the embedding $E^{\\mathrm{geo}}_{[p]}(W,\\mu)(\\overline{\\gamma})^* \\colon \\mathrm{lab}_{\\alpha}(W,\\mu)|^{\\gamma^{-1} \\llparenthesis\\hspace{.1em}\\mathring{q}'\\hspace{.1em}\\rrparenthesis}\\hookrightarrow \\mathrm{lab}_{\\alpha'}(W,\\mu)$ is an isotopy equivalence over $\\gamma$. To see this, note that since $\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q}'\\hspace{.1em}\\rrparenthesis\\rightarrow \\llparenthesis\\hspace{.1em} \\mathring{q}'\\hspace{.1em}\\rrparenthesis$ is bijective, the embedding $\\overline{\\gamma}\\colon \\mathrm{wlab}_\\alpha(\\ensuremath{\\mathbf{R}})|^{\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q}'\\hspace{.1em}\\rrparenthesis}\\hookrightarrow \\mathrm{wlab}_{\\alpha'}(\\ensuremath{\\mathbf{R}})$ is an isotopy equivalence over $\\gamma$ and under $\\mathrm{coll}(\\ensuremath{\\mathbf{R}})|^{\\alpha'^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q}'\\hspace{.1em}\\rrparenthesis}$, from which it follows that $E^{\\mathrm{geo}}_{[p]}(W,\\mu)(\\overline{\\gamma})$ is an isotopy equivalence over $\\gamma$ as claimed.\n\\end{proof}\n\nBy the previous lemma, \\eqref{equ:bord-to-pre-morita-nonalg} restricts to a morphism $\\gls*{psi} \\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\rightarrow\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)$ of semisimplicial $\\infty$-categories. This completes \\ref{step:functor-to-premorita}.\n\n\\subsection{Composite algebras}\\label{step:composite}\nWe now consider the composition\n\\begin{equation}\n\t\\label{equ:non-unital-composition}\\gls*{overlinee} \\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\xrightarrow{E^{\\mathrm{geo}}}\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)\\xlra{y_*}\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d))\\xlra{\\iota^*}\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)).\n\\end{equation}\nHere $E^{\\mathrm{geo}}$ is the functor from the previous step, $y_*$ is induced by the (monoidal) Yoneda embedding $y\\colon \\ensuremath{\\icat{M}\\mathrm{an}}_d\\rightarrow \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)$ (see \\cref{sec:presheaf-category}), and $\\iota^*$ is the functor induced by the lax monoidal functor $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)\\rightarrow \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ which is itself induced by the inclusion $\\iota\\colon \\ensuremath{\\icat{D}\\mathrm{isc}}_d\\hookrightarrow \\ensuremath{\\icat{M}\\mathrm{an}}_d$ of the full subcategory spanned by manifolds diffeomorphic to $T \\times \\ensuremath{\\mathbf{R}}^d$ for finite sets $T$ with monoidal structure inherited from $\\ensuremath{\\icat{M}\\mathrm{an}}$. By the properties of presheaf categories discussed in \\cref{sec:presheaf-category}, the monoidal category $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ has good relative tensor products in the sense of  \\cref{sec:composite-algebras}, so it makes sense to ask whether \\eqref{equ:non-unital-composition} lands in the levelwise full subcategory $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))\\subset \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))$\nof \\cref{sec:composite-algebras}. This section serves to prove this:\n\n\\begin{prop}\\label{prop:image-in-Morita}\nThe functor $\\smash{\\overline{E}}$ from \\eqref{equ:non-unital-composition} factors through $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))\\subset \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))$.\n\\end{prop}\n\nWe will first explain how \\cref{prop:image-in-Morita} follows from a seemingly different result, and then prove that other result.  The argument involves a simplicial thickening\n\\[\n\t\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}^\\rhd}\\xlra{\\simeq}\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^{\\rhd}.\n\\]\nof the right-cone $\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^{\\rhd}$ of the category $\\ensuremath{\\cat{Cut}}_\\mathrm{sur}$ (the category obtained by freely adding a terminal object $\\infty\\in\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^{\\rhd}$) in terms of the manifolds \n\\[\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis^*\\coloneqq L \\times [-\\epsilon,\\epsilon) \\cup \\llparenthesis\\hspace{.1em} \\mathring{a} \\hspace{.1em}\\rrparenthesis \\times (-\\epsilon,\\epsilon) \\cup R \\times (-\\epsilon,\\epsilon] \\subset \\llparenthesis\\hspace{.1em} a \\hspace{.1em}\\rrparenthesis \\times \\ensuremath{\\mathbf{R}} \\quad\\text{and} \\quad\t\\llparenthesis\\hspace{.1em} \\infty\\hspace{.1em}\\rrparenthesis^*\\coloneqq [-\\epsilon,\\epsilon]\\subset\\ensuremath{\\mathbf{R}}\\]\nwhere $a\\ge0$. The objects of $\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^{\\rhd}}$ are the same as those of $\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^{\\rhd}$. The space of morphisms $\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis\\rightarrow \\llparenthesis\\hspace{.1em} b\\hspace{.1em}\\rrparenthesis$ between objects of $\\ensuremath{\\cat{Cut}}^\\mathrm{sur}\\subset \\ensuremath{\\cat{Cut}}_\\mathrm{sur}^{\\rhd}$ is defined as\n\\[\\textstyle{\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}^\\rhd}}(\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis, \\llparenthesis\\hspace{.1em} b\\hspace{.1em}\\rrparenthesis)\\coloneqq \\bigsqcup_{\\gamma\\in \\mathrm{Map}_\\ensuremath{\\cat{Cut}}(\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} b\\hspace{.1em}\\rrparenthesis)} \\ensuremath{\\mathrm{Emb}}\\big(\\llparenthesis\\hspace{.1em} a \\hspace{.1em}\\rrparenthesis^\\ast,\\llparenthesis\\hspace{.1em} b \\hspace{.1em}\\rrparenthesis^\\ast\\big)_{\\gamma}}\n\\]\nwhere the subscript $(-)_{\\gamma}$ indicates we restrict to embeddings $\\overline{\\gamma}$ that cover $\\gamma$, are the identity on $L \\times [-\\epsilon,-\\tfrac{\\epsilon}{2}) \\cup R \\times (\\tfrac{\\epsilon}{2},\\epsilon]$ and preserve the lexicographic order inherited from $\\llparenthesis\\hspace{.1em} a \\hspace{.1em}\\rrparenthesis \\times \\ensuremath{\\mathbf{R}}$ and $\\llparenthesis\\hspace{.1em} b \\hspace{.1em}\\rrparenthesis \\times \\ensuremath{\\mathbf{R}}$. Finally, the space of morphisms $\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis \\rightarrow \\llparenthesis\\hspace{.1em} \\infty\\hspace{.1em}\\rrparenthesis$ is defined as\n\\[\n\t\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}^\\rhd}}(\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis, \\llparenthesis\\hspace{.1em} \\infty\\hspace{.1em}\\rrparenthesis)\\coloneq\n\\ensuremath{\\mathrm{Emb}}\\big(\\llparenthesis\\hspace{.1em} a \\hspace{.1em}\\rrparenthesis^\\ast,\\llparenthesis\\hspace{.1em} \\infty\\hspace{.1em}\\rrparenthesis^*\\big)_{\\infty}\n\\]\nwhere the subscript $(-)_\\infty$ indicates that we restrict to embeddings $\\overline{\\gamma}$ that agree on $L \\times [-\\epsilon,-\\tfrac{\\epsilon}{2}) \\cup R \\times (\\tfrac{\\epsilon}{2},\\epsilon]$ with the projection to the second coordinate and preserve the lexicographical order inherited from $\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}$ and $\\ensuremath{\\mathbf{R}}$. The space of morphisms $\\llparenthesis\\hspace{.1em}\\infty\\hspace{.1em}\\rrparenthesis \\rightarrow \\llparenthesis\\hspace{.1em}\\infty\\hspace{.1em}\\rrparenthesis$ is the space of self-embeddings of $\\llparenthesis\\hspace{.1em}\\infty\\hspace{.1em}\\rrparenthesis^\\ast=[-\\epsilon,\\epsilon]$ that agree with the identity on the complement of $[-\\tfrac{\\epsilon}{2},\\tfrac{\\epsilon}{2}]$. This category admits an evident functor to $\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^{\\rhd}$ which is an equivalence as a result of the contractibility of the space of order-preserving embeddings between intervals.\n\n\\begin{nconvention}In what followsm we occasionally omit the choices of embeddings of manifolds into Euclidean spaces for brevity. For instance, we treat $\\ensuremath{\\cat{Man}}_d=(\\ensuremath{\\cat{Man}}_d^{\\sqcup})_{[1]}$ from \\cref{step:mand} as the $\\ensuremath{\\cat{Kan}}$-enriched category of abstract smooth $d$-manifolds and codimension $0$ embeddings.\n\\end{nconvention}\n\nGiven a (possibly noncompact) $d$-manifold without boundary $V$  equipped with $k$ disjoint codimension $1$ submanifolds $V_i\\subset V$ that are topologically closed in $V$ as a subspace, equipped with disjoint bicollars $[-\\epsilon,\\epsilon]\\times V_i\\subset V$, we construct a simplicially enriched functor\n\\[\n\tV{\\llparenthesis\\hspace{.1em} -\\hspace{.1em}\\rrparenthesis}\\colon \\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}\\longrightarrow \\ensuremath{\\cat{Man}}_d\n\\]\nwhich on objects sends $\\llparenthesis\\hspace{.1em} \\infty\\hspace{.1em}\\rrparenthesis $ to $V{\\llparenthesis\\hspace{.1em} \\infty\\hspace{.1em}\\rrparenthesis}\\coloneqq V$ and $ \\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis\\in \\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}}$ to\n\\[\n\\textstyle{V{\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis}\\coloneqq V^\\ast\\sqcup\\big( \\bigsqcup_{i=1}^k\\llparenthesis\\hspace{.1em} \\mathring{a}\\hspace{.1em}\\rrparenthesis\\times(-\\epsilon,\\epsilon)\\times V_i}\\big),\n\\]\nwhere $V^\\ast$ is the manifold obtained from $V$ by cutting out $\\cup_{i=1}^k[-\\tfrac{\\epsilon}{2},\\tfrac{\\epsilon}{2}]\\times V_i$ and extending the resulting collars $[-\\epsilon,-\\tfrac{\\epsilon}{2})\\times V_i\\sqcup (\\tfrac{\\epsilon}{2},\\epsilon]\\times V_i$ to collars $[-\\epsilon,\\epsilon)\\times V_i\\sqcup (-\\epsilon,\\epsilon]\\times V_i$. Given a morphism $\\overline{\\gamma} \\colon \\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis \\to \\llparenthesis\\hspace{.1em} b \\hspace{.1em}\\rrparenthesis$ there is an embedding $V{\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis} \\hookrightarrow V{\\llparenthesis\\hspace{.1em} b\\hspace{.1em}\\rrparenthesis}$ that is the identity of $V^\\ast$ outside the extended collars, and agrees on the remaining part with $\\overline{\\gamma} \\times \\mathrm{id}_{V_i}$. Finally, for $\\llparenthesis\\hspace{.1em} a \\hspace{.1em}\\rrparenthesis \\to \\llparenthesis\\hspace{.1em}\\infty\\hspace{.1em}\\rrparenthesis$ or $\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis \\to \\llparenthesis\\hspace{.1em}\\infty\\hspace{.1em}\\rrparenthesis$ one defines embeddings $V{\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis} \\hookrightarrow V{\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis}$ or  $V{\\llparenthesis\\hspace{.1em} \\infty\\hspace{.1em}\\rrparenthesis} \\hookrightarrow V{\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis}$ in the same manner. \n\nWriting $\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}}\\subset \\underline{\\ensuremath{\\cat{Cut}}^\\rhd_\\mathrm{sur}}$ for the full subcategory covering the inclusion $\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}\\subset\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}^{\\rhd}$, \\cref{prop:image-in-Morita} will be a consequence of the following proposition involving homotopy colimits in the Kan--Quillen model structure on $\\cat{S}$.\n\n\\begin{prop}\\label{prop:cutting-hocolim}For a manifold $D$ diffeomorphic to $T\\times \\ensuremath{\\mathbf{R}}^d$ for a finite set $T$, the map\n\\[\n\t\\mathrm{hocolim}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}}}\\,\\ensuremath{\\mathrm{Emb}}(D,V{\\llparenthesis\\hspace{.1em}-\\hspace{.1em}\\rrparenthesis})\\longrightarrow \\mathrm{hocolim}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}\\,\\ensuremath{\\mathrm{Emb}}(D,V{\\llparenthesis\\hspace{.1em}-\\hspace{.1em}\\rrparenthesis})\\simeq \\ensuremath{\\mathrm{Emb}}(D,V{\\llparenthesis\\hspace{.1em}\\infty\\hspace{.1em}\\rrparenthesis})\n\\]\ninduced by the inclusion $\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}} \\subset \\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}$ is an equivalence.\n\\end{prop}\n\nWe postpone the proof to the next subsection and first explain how it implies  \\cref{prop:image-in-Morita}.\n\n\\begin{proof}[Proof of \\cref{prop:image-in-Morita}]\nConsulting the definition of the Morita category, we have to show that the image of any object $\\smash{(W,\\mu)\\in\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}_{[p]}}$ in $\\smash{\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))_{[p]}}$ is composite in the sense of \\cref{sec:composite-algebras}. By \\cref{cor:composite-algebras-multisimplicial} this is equivalent to proving that for each  $\\alpha\\in\\Delta^{\\mathrm{op}}_{\/[p]}$ \n\\vspace{-0.1cm}\n\\begin{equation}\\label{equ:composition-test-composite}\n\t(\\Delta^\\mathrm{op}_\\mathrm{inj})^\\rhd\\xra{\\eta^\\alpha} \\Delta^{\\mathrm{act},\\mathrm{op}}_{\/[p]} \\xra{E^{\\mathrm{geo}}_{[p]}(W,\\mu)} \\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup,\\mathrm{act}}\\xra{(-)_!}\\ensuremath{\\icat{M}\\mathrm{an}}_d\n\\end{equation}\nbecomes a colimit diagram when postcomposed with $(\\iota^*\\circ y)\\colon \\ensuremath{\\icat{M}\\mathrm{an}}_d\\rightarrow\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$. We first make the composition \\eqref{equ:composition-test-composite} more explicit. Recall from \\ref{step:functor-to-premorita} \\ref{step:functor-to-premorita-simplicial} that \\[E^{\\mathrm{geo}}_{[p]}(W,\\mu)\\in\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)_{[p]}\\subset \\ensuremath{\\mathrm{Fun}}_{\\Delta^{\\mathrm{op}}}(\\Delta^{\\mathrm{op}}_{\/[p]},\\ensuremath{\\icat{M}\\mathrm{an}}^{\\sqcup}_d)\\] was obtained from a functor between simplicially enriched categories \\begin{equation}\\label{equ:psiW-simplicial}\n\tE^{\\mathrm{geo}}_{[p]}(W,\\mu)\\colon \\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\/}\\rightarrow \\ensuremath{\\cat{Man}}^{\\sqcup}_d\n\\end{equation} \nby taking coherent nerves and using the equivalence $\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\/}\\simeq \\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\/}\\cong \\Delta^{\\mathrm{op}}_{\/[p]}$ from \\ref{step:functor-to-premorita} \\ref{step:functor-to-premorita-thickening}. We now give a similar description of the composition \\eqref{equ:composition-test-composite} as a simplicially enriched functor using a simplicial functor to the full subcategory $\\underline{\\ensuremath{\\cat{Cut}}}^\\mathrm{act}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}\\subset \\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}$ covering $\\ensuremath{\\cat{Cut}}^\\mathrm{act}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}\\subset \\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}$\n \\[\\underline{\\eta^\\alpha}\\colon \\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}\\longrightarrow\\underline{\\ensuremath{\\cat{Cut}}}^\\mathrm{act}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}\\] to the pullback $\\underline{\\ensuremath{\\cat{Cut}}}^\\mathrm{act}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}$ of $\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}$ along $\\ensuremath{\\cat{Cut}}^\\mathrm{act}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}\\subset \\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}$. The functor $\\underline{\\eta^\\alpha}$ will make \n\\[\\begin{tikzcd}\n\t\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}\\dar{\\simeq}\\arrow[r,\"\\underline{\\eta^\\alpha}\"]&\\underline{\\ensuremath{\\cat{Cut}}}^\\mathrm{act}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}\\dar{\\simeq}\\\\[-2pt]\n\t\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd\\rar{\\eta^\\alpha}&\\ensuremath{\\cat{Cut}}^\\mathrm{act}_{\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis\/}.\n\\end{tikzcd}\\]\ncommutative where $\\eta^\\alpha$ is the functor from \\cref{sec:composite-multisimplicial}. The construction involves the notation of \\cref{const:rhoalpha-cut} ($k_\\alpha$, $\\alpha_1^{\\vec{a}}$, $n_i$, etc.) and the discussion preceding \\cref{cor:composite-algebras-multisimplicial}. On objects, $\\smash{\\underline{\\eta^\\alpha}}$ is determined by $\\eta^\\alpha$. On morphisms it sends $\\overline{\\gamma}\\colon \\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis^\\ast\\hookrightarrow  \\llparenthesis\\hspace{.1em} b\\hspace{.1em}\\rrparenthesis^\\ast$ to the right-hand embedding in a commutative square of embeddings (here $\\vec{a}=(a,\\ldots,a)$ and $\\vec{b}=(b,\\ldots,b)$)\n\\[\\begin{tikzcd}[ar symbol\/.style = {draw=none,\"\\textstyle#1\" description,sloped},\tsubset\/.style = {ar symbol={\\subset}}, supset\/.style = {ar symbol={\\supset}}]\n\t\\sqcup^{k_\\alpha} \\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis\\times \\ensuremath{\\mathbf{R}}&[-15pt] \\arrow[l,supset]\\sqcup^{k_\\alpha}\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis^\\ast\\rar[r,hookrightarrow]\\arrow[d,hookrightarrow,\"{\\sqcup^{k_\\alpha}\\overline{\\gamma}}\",swap]& \\mathrm{wlab}_{\\alpha_1^{\\vec{a}}}(\\ensuremath{\\mathbf{R}})\\arrow[d,hookrightarrow]\\arrow[r,subset]& [-15pt]\\llparenthesis\\hspace{.1em} k_{\\alpha}^{\\vec{a}}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\\\\\n\t\\sqcup^{k_\\alpha} \\llparenthesis\\hspace{.1em} b\\hspace{.1em}\\rrparenthesis\\times \\ensuremath{\\mathbf{R}}& \\arrow[l,supset] \\sqcup^{k_\\alpha}\\llparenthesis\\hspace{.1em} b\\hspace{.1em}\\rrparenthesis^\\ast\\arrow[r,hookrightarrow]& \\mathrm{wlab}_{\\alpha_1^{\\vec{b}}}(\\ensuremath{\\mathbf{R}})\\arrow[r,subset]&\\llparenthesis\\hspace{.1em} k_{\\alpha}^{\\vec{b}}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}.\n\\end{tikzcd}\\]\nThe $i$th component of the upper horizontal map is the embedding \\[\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}} \\supset \\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis^*\\lhook\\joinrel\\longrightarrow \\mathrm{wlab}_{\\alpha_1^{\\vec{a}}}(\\ensuremath{\\mathbf{R}})\\subset\\llparenthesis\\hspace{.1em} k_{\\alpha}^{\\vec{a}}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\\] that is the unique inclusion of components that preserves the lexicographic order inherited from $\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}$ and $\\llparenthesis\\hspace{.1em} k_{\\alpha}^{\\vec{a}}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}$ and covers the map $\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis\\rightarrow \\llparenthesis\\hspace{.1em} k_{\\alpha}^{\\vec{a}}\\hspace{.1em}\\rrparenthesis$ given by the sequence $\\alpha_1^{\\vec{a}}(n_i)<\\alpha_1^{\\vec{a}}(n_i)+1<\\ldots <\\alpha_1^{\\vec{a}}(n_i)+a<\\alpha_1^{\\vec{a}}(n_i+1)$ (note that this is \\emph{not} a morphism in $\\ensuremath{\\cat{Cut}}$ as it does not preserve the endpoints). The bottom horizontal embedding is defined in the same way, and the right hand embedding is defined to agree with $\\overline{\\gamma}$ on the components hit by the horizontal embedding and on the complement as the unique inclusion of components that covers the map $\\eta^\\alpha(\\gamma)\\colon \\llparenthesis\\hspace{.1em} k_{\\alpha}^{\\vec{a}}\\hspace{.1em}\\rrparenthesis \\rightarrow \\llparenthesis\\hspace{.1em} k_{\\alpha}^{\\vec{b}}\\hspace{.1em}\\rrparenthesis $ and preserves the lexicographic order. Similarly, $\\underline{\\eta}^\\alpha$ sends a morphism in $\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}$ given by an embedding $\\overline{\\gamma}\\colon \\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis^\\ast\\hookrightarrow \\llparenthesis\\hspace{.1em} \\infty\\hspace{.1em}\\rrparenthesis^\\ast$ to the right-hand embedding in the square\n\\[\\begin{tikzcd}[ar symbol\/.style = {draw=none,\"\\textstyle#1\" description,sloped},\n\t\tsubset\/.style = {ar symbol={\\subset}}, supset\/.style = {ar symbol={\\supset}}]\n\t\t\\sqcup^{k_\\alpha} \\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis\\times \\ensuremath{\\mathbf{R}}&[-15pt]\\arrow[l,supset]\\sqcup^{k_\\alpha}\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis^\\ast\\rar[r,hookrightarrow]\\arrow[d,hookrightarrow,\"{\\sqcup^{k_\\alpha}\\overline{\\gamma}}\",swap]&\\mathrm{wlab}_{\\alpha_1^{\\vec{a}}}(\\ensuremath{\\mathbf{R}})\\arrow[d,hookrightarrow]\\arrow[r,subset]&[-15pt]\\llparenthesis\\hspace{.1em} k_{\\alpha}^{\\vec{a}}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\\\\\n\t\t& \\sqcup^{k_\\alpha}\\llparenthesis\\hspace{.1em} \\infty^\\ast\\hspace{.1em}\\rrparenthesis\\arrow[r,hookrightarrow]&\\mathrm{wlab}_{\\alpha}(\\ensuremath{\\mathbf{R}})\\arrow[r,subset]&\\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\n\\end{tikzcd}\\]\nwhere the top horizontal embedding is the same as before, the bottom embedding includes the $i$th copy of $\\llparenthesis\\hspace{.1em} \\infty^\\ast\\hspace{.1em}\\rrparenthesis=[-\\epsilon,\\epsilon]$ as the unique $[-\\epsilon,\\epsilon]$-component in $\\mathrm{wlab}_{\\alpha}(\\ensuremath{\\mathbf{R}})$ that maps to $\\alpha(n_i)\\in \\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis$ under the projection (using the notation from \\cref{const:rhoalpha-cut}) and to $n_i\\in\\llparenthesis\\hspace{.1em} \\mathring{p-1}\\hspace{.1em}\\rrparenthesis=\\{1,\\ldots,p-1\\}\\subset[p]$ under the map $\\mathrm{wlab}_{\\alpha}(\\ensuremath{\\mathbf{R}})\\rightarrow [p]$ from \\ref{step:functor-to-premorita} \\ref{step:functor-to-premorita-language} II. The right vertical embedding is defined via the left vertical one on the components hit by the horizontal map and as the unique inclusion of components that cover the map $\\gamma_{\\vec{a}}\\colon \\llparenthesis\\hspace{.1em} k_{\\alpha}^{\\vec{a}}\\hspace{.1em}\\rrparenthesis\\rightarrow \\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis$ and preserve the lexicographic order on $\\llparenthesis\\hspace{.1em} k_{\\alpha}^{\\vec{a}}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}$ and $\\llparenthesis\\hspace{.1em} q\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}$.\n\t\nBy construction the composition \\eqref{equ:composition-test-composite} is equivalent to the coherent nerve of the composition\n\\vspace{-0.1cm}\n\\begin{equation}\\label{equ:colimit-diagram-psiW-simplicial-n}\n\t\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}\\xlra{\\underline{\\eta^\\alpha}} \\underline{\\ensuremath{\\cat{Cut}}}^{\\mathrm{act}}_{\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\/} \\xrightarrow{E^{\\mathrm{geo}}_p(W,\\mu)} \\ensuremath{\\cat{Man}}_d^{\\sqcup,\\mathrm{act}}\\xlra{(-)_!}\\ensuremath{\\cat{Man}}_d\n\\end{equation}\nwhere $(-)_!$ is the simplicial ``disjoint unions''-functor of \\eqref{equ:active-pushforward}. Tracing through the definitions, one checks that this functor agrees up to equivalence with the functor $V\\llparenthesis\\hspace{.1em} -\\hspace{.1em}\\rrparenthesis$ for the manifold $V=\\mathrm{lab}_\\alpha(W,\\mu)_!$ with the $k_\\alpha$ different bicollared submanifolds $[-\\epsilon,\\epsilon]\\times W_{\\mu(j)}\\cong W_{[\\mu(j)-\\epsilon,\\mu(j)+\\epsilon]}\\subset \\mathrm{lab}_\\alpha(W,\\mu)_!$ for $j\\in\\llparenthesis\\hspace{.1em} \\mathring{p-1}\\hspace{.1em}\\rrparenthesis$ with $\\alpha(j)\\in\\llparenthesis\\hspace{.1em} \\mathring{q}\\hspace{.1em}\\rrparenthesis$ and $ \\alpha(j)=\\alpha(j+1)$. Using that a diagram $A\\colon K^{\\rhd}\\rightarrow \\ensuremath{\\catsingle{C}}$ is a colimit diagram if and only if the natural map $\\mathrm{colim}_{K}A\\rightarrow \\mathrm{colim}_{K^{\\rhd}}A$ is an equivalence, this implies that it suffices to show that the colimit\n\\[\n\t\\mathrm{colim}_{N_{\\mathrm{coh}}(\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd})}\\Big(N_{\\mathrm{coh}}(\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd})) \\xra{N_{\\mathrm{coh}} (V\\llparenthesis\\hspace{.1em}-\\hspace{.1em}\\rrparenthesis)}N_{\\mathrm{coh}}(\\ensuremath{\\cat{Man}}_d) \\xra{y} \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)\\xra{\\iota^*}\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)\\Big)\n\\]\nis unaffected by precomposing the diagram with the functor $N_{\\mathrm{coh}}(\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}})\\rightarrow N_{\\mathrm{coh}}(\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd})$ induced by inclusion. Using that (i) equivalences in functor categories are detected objectwise, (ii) colimits in functor categories commute with evaluation at a fixed object $D\\in\\ensuremath{\\icat{D}\\mathrm{isc}}_d$ \\cite[5.1.2.3]{LurieHTT}, and (iii) the compatibility of the simplicial and $\\infty$-categorical Yoneda embedding (see \\cref{fact:yoneda-comparison}), we see that it is enough to show that the colimit\n\\[\n\t\\mathrm{colim}_{N_{\\mathrm{coh}}(\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd})}\\Big(N_{\\mathrm{coh}}(\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd})\\xra{N_{\\mathrm{coh}}(\\mathrm{ev}_D\\circ y_s\\circ V\\llparenthesis\\hspace{.1em} -\\hspace{.1em}\\rrparenthesis)}N_{\\mathrm{coh}}(\\ensuremath{\\cat{Kan}})\\Big)\n\\]\nis unaffected by precomposing the diagram with $N_{\\mathrm{coh}}(\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}})\\rightarrow N_{\\mathrm{coh}}(\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd})$ for each object $D\\in\\ensuremath{\\icat{D}\\mathrm{isc}}_d$ where $y_s\\colon \\ensuremath{\\cat{Man}}_d\\rightarrow\\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{Man}}_d,\\ensuremath{\\cat{Kan}})$ is the simplicial Yoneda embedding of the $\\ensuremath{\\cat{Kan}}$-enriched category $\\ensuremath{\\cat{Man}}_d$.  Using that model category-theoretic homotopy colimits are compatible with $\\infty$-categorical colimits \\cite[4.2.4.1]{LurieHTT}, the claim reduces to showing that the natural map between homotopy colimits in the Kan--Quillen model structure\n\\[\n\t\\mathrm{hocolim}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}}}\\big(\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}}\\xra{\\mathrm{ev}_D\\circ y_s\\circ V\\llparenthesis\\hspace{.1em} -\\hspace{.1em}\\rrparenthesis}\\ensuremath{\\cat{S}}\\big)\\rightarrow\n\t\\mathrm{hocolim}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}\\big(\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}\\xra{\\mathrm{ev}_D\\circ y_s\\circ V\\llparenthesis\\hspace{.1em} -\\hspace{.1em}\\rrparenthesis}\\ensuremath{\\cat{S}}\\big)\n\\]\nis an equivalence. This is \\cref{prop:cutting-hocolim}.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{prop:cutting-hocolim}]This proof will eventually rely on a microfibration argument, which is why we phrase the argument in the category of topological spaces $\\ensuremath{\\cat{Top}}$ as opposed to simplicial sets $\\ensuremath{\\cat{S}}$. Relying on the usual Quillen equivalence between the category of simplicial sets $\\ensuremath{\\cat{S}}$ and that of topological spaces $\\ensuremath{\\cat{Top}}$, the claim has an evident reformulation in terms of homotopy colimits of $\\ensuremath{\\cat{Top}}$-enriched $\\ensuremath{\\cat{Top}}$-valued functors and it is this reformulation that we shall prove.\n\t\nTo begin with, we note that it suffices to show the claim for $D=\\underline{n}\\times\\ensuremath{\\mathbf{R}}^d$ for $n\\ge0$. Next, we simplify the functor $\\ensuremath{\\mathrm{Emb}}(\\underline{n}\\times\\ensuremath{\\mathbf{R}},-)\\colon \\ensuremath{\\cat{Man}}_d\\rightarrow\\ensuremath{\\cat{Top}}$ in terms of the functor $C_n^{\\mathrm{fr}}\\colon\\ensuremath{\\cat{Man}}_d\\rightarrow\\ensuremath{\\cat{Top}}$ given by taking framed configurations, i.e.\\,the pullback of functors\n\\[\\begin{tikzcd} \n\tC^\\mathrm{fr}_n(-) \\rar \\dar &[-2pt] \\mathrm{Map}(\\ul{n},\\mathrm{Fr}(-)) \\dar \\\\[-2pt]\n\t\\ensuremath{\\mathrm{Emb}}(\\ul{n},-) \\rar{\\subset} & \\mathrm{Map}(\\ul{n},-)\n\\end{tikzcd}\\]\nwhose right vertical map is induced by the projection $\\mathrm{Fr}(W)\\rightarrow W$ of the frame bundle of manifolds $W\\in\\ensuremath{\\cat{Man}}_d$. Taking derivatives at the centres $\\underline{n}\\times\\{0\\}\\subset \\underline{n}\\times\\ensuremath{\\mathbf{R}}^d$ gives a natural transformation $\\ensuremath{\\mathrm{Emb}}(\\underline{n}\\times\\ensuremath{\\mathbf{R}},-)\\rightarrow C^\\mathrm{fr}_n(-)$ which is a componentwise weak equivalence, so we conclude that in order to prove \\cref{prop:cutting-hocolim} it suffices to show that the map\n\\[\n\t\\mathrm{hocolim}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}}}\\big(C_n^{\\mathrm{fr}}(V\\llparenthesis\\hspace{.1em}-\\hspace{.1em}\\rrparenthesis)\\big)\\longrightarrow\n\t\\mathrm{hocolim}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}\\big(C_n^{\\mathrm{fr}}(V\\llparenthesis\\hspace{.1em}-\\hspace{.1em}\\rrparenthesis)\\big)\n\\]\nis a weak equivalence. This is a map between homotopy colimits in spaces, which we model by a bar construction. In general, given a $\\ensuremath{\\cat{Top}}$-enriched category $\\cat{C}$ and $\\ensuremath{\\cat{Top}}$-enriched functors $F \\colon \\cat{C} \\to \\ensuremath{\\cat{Top}}$ and $G \\colon \\cat{C}^\\mathrm{op} \\to \\ensuremath{\\cat{Top}}$, the \\emph{bar-construction} $B_\\bullet(F,\\cat{C},G) \\colon \\Delta^\\mathrm{op} \\rightarrow \\ensuremath{\\cat{Top}}$ is the simplicial space $\\textstyle{[r] \\longmapsto \\bigsqcup_{(c_0,\\ldots,c_r)} F(c_0) \\times \\cat{C}(c_0,c_1) \\times \\cdots \\times \\cat{C}(c_{r-1},c_r) \\times G(c_r)}$ where $(c_0,\\ldots,c_r)$ tuns through ordered sequences of $(r+1)$ objects in $\\cat{C}$. If $G$ has weakly contractible values, the thick geometric realisation $B(F,\\cat{C},G)\\coloneqq \\|B_\\bullet(F,\\cat{C},G)\\|$ is a model for $\\mathrm{hocolim}_\\cat{C} F$ (see e.g.\\,\\cite[Corollary 9.2.7]{Riehl}; since we take thick geometric realisations we do not need to worry about cofibrancy issues). Choosing $\\cat{C}=\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}$ and $G=\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(-,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)$, it therefore suffices to that\n\\begin{equation}\\label{equ:simplicial-composite-version}\n\tB_\\bullet\\big(C^\\mathrm{fr}_n,\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}},\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(-,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)\\big) \\longrightarrow B_\\bullet\\big(C^\\mathrm{fr}_n,\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd},\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(-,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)\\big) \n\\end{equation}\ninduced by $\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}}\\subset \\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}$ is a weak equivalence on thick realisations. There is an augmentation\n\\begin{equation}\\label{equ:bar-resolution}\n\tB_\\bullet\\big(C^\\mathrm{fr}_n,\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd},\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(-,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)\\big) \\longrightarrow C^\\mathrm{fr}_n(V)\n\\end{equation} \ninduced by composition of embeddings and evaluation of $C_n^{\\mathrm{fr}}(-)$. This admits an \\emph{extra degeneracy} so it induces an equivalence on (thick) realisation (see e.g.\\,\\cite[Example 4.5.7]{Riehl}). This leaves us with showing that the composition of \\eqref{equ:simplicial-composite-version} and \\eqref{equ:bar-resolution}\n\\begin{equation}\\label{equ:augmentation-of-bar}\n\tB_\\bullet\\big(C^\\mathrm{fr}_n,\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}},\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(-,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)\\big) \\longrightarrow C^\\mathrm{fr}_n(V)\n\\end{equation}  is an equivalence on thick realisations. To prove this, we consider a semisimplicial space $\\mathrm{wall}_\\bullet$ whose space of $p$-simplices is the space of order-preserving functions $\\tau \\colon [p] \\to (-\\epsilon,\\epsilon)$ with simplicial structure by precomposition, and we define an augmented semisimplicial space\n\\begin{equation}\\label{equ:resolve-embeddings-into-infty}\n\t\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)_\\bullet\\longrightarrow \\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)\n\\end{equation}\n for $a\\ge0$ whose space of $p$-simplices \n\\begin{equation}\\label{equ:thicken-maps-to-infty}\n\t\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)_p\\subset \\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)\\times \\mathrm{wall}_p\n\\end{equation} \nis the subspace of pairs of a function $\\tau\\colon[p]\\rightarrow (\\epsilon,\\epsilon)$ and an embedding $\\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis^* \\hookrightarrow\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis^*$ that is disjoint from the image of $\\tau$. Varying $a$, this defines a functor $\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\mathrm{op}\\times\\Delta_\\mathrm{inj}^{\\mathrm{op}}\\longrightarrow \\ensuremath{\\cat{Top}}$ that is compatible with \\eqref{equ:resolve-embeddings-into-infty}, so we obtain an augmentation\n\\begin{equation}\\label{equ:map-on-bar}\n\tB\\big(C^\\mathrm{fr}_n,\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}},\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(-,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)_\\bullet\\big)\\longrightarrow B\\big(C^\\mathrm{fr}_n,\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}},\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(-,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)\\big).\n\\end{equation}\nIn \\cref{lem:microfibration-lemma} \\ref{enum:microfibration-lemma-i} below, we will show that \\eqref{equ:resolve-embeddings-into-infty} realises to a weak equivalence. Together with the fact that, up to weak equivalence, it does not matter in which direction one realises a semisimplicial space first, this implies that the map in \\eqref{equ:map-on-bar} realises to a weak equivalence, so it remains to show that the augmented semisimplicial space \n\\[\n\tB\\big(C^\\mathrm{fr}_n,\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}},\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(-,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)_\\bullet\\big)\\longrightarrow C_n^\\mathrm{fr}(V)\n\\]\nobtained by combining \\eqref{equ:map-on-bar} and \\eqref{equ:augmentation-of-bar} realises to a weak equivalence. To prove this remaining claim, we consider the sub-simplicial space $\\mathrm{wall}^V_\\bullet\\subset \\mathrm{wall}_\\bullet\\times C_n^{\\mathrm{fr}}(V)$ consisting of pairs of a function $\\tau\\colon [p]\\rightarrow(-\\epsilon,\\epsilon)$ and a framed configurations $\\vec{x}\\in C_n(V)$ that is disjoint from the submanifolds $\\{\\tau(j)\\}\\times V_i\\subset V$ for all $j=0,\\ldots,p$ and $i=1,\\ldots,k$ (here we used the collars $[-\\epsilon,\\epsilon]\\times V_i\\subset V$; see \\cref{fig:wall-semi} for an example. The projection to $\\mathrm{wall}_p$ in \\eqref{equ:thicken-maps-to-infty} and the augmentation to $C_n^\\mathrm{fr}(V)$ assemble to a semisimplicial map over $C_n^\\mathrm{fr}(V)$\n\\begin{equation}\\label{equ:bar-to-wall}\n\tB\\big(C^\\mathrm{fr}_n,\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}},\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(-,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)_\\bullet\\big)\\longrightarrow \\mathrm{wall}^V_\\bullet\n\\end{equation}\nwhich we show to be a levelwise weak equivalence in \\cref{lem:microfibration-lemma} \\ref{enum:microfibration-lemma-ii}. This leaves us with showing that the augmentation $\\mathrm{wall}_\\bullet^V\\rightarrow C_n^{\\mathrm{fr}}(V)$ realises to a weak equivalence. This is \\cref{lem:microfibration-lemma} \\ref{enum:microfibration-lemma-iii}.\n\\end{proof}\n\n\\begin{figure}\n\t\t\\begin{tikzpicture}\n\t\t\\draw [dashed] (-3.3,3.5) -- (-3.3,0);\n\t\t\\draw [dashed] (-2.5,3.5) -- (-2.5,0);\n\t\t\\draw [dashed] (-1.9,3.5) -- (-1.9,0);\n\t\t\\draw [|-|] (-3.5,0) -- (-1.5,0);\n\t\t\\node at (-3.3,-.3) {\\tiny $\\tau(0)$};\n\t\t\\node at (-2.5,-.3) {\\tiny $\\tau(1)$};\n\t\t\\node at (-1.9,-.3) {\\tiny $\\tau(2)$};\n\t\t\n\t\t\\draw [dashed] (1.2,3.5) -- (1.2,0);\n\t\t\\draw [dashed] (2,3.5) -- (2,0);\n\t\t\\draw [dashed] (2.6,3.5) -- (2.6,0);\n\t\t\\draw [|-|] (1,0) -- (3,0);\n\t\t\\node at (1.2,-.3) {\\tiny $\\tau(0)$};\n\t\t\\node at (2,-.3) {\\tiny $\\tau(1)$};\n\t\t\\node at (2.6,-.3) {\\tiny $\\tau(2)$};\n\t\t\n\t\t\\draw [dotted,Mahogany] (-5.5,3.5) -- (-5,3);\n\t\t\\draw [dotted,Mahogany] (-5.5,2.5) -- (-5,2);\n\t\t\\draw [dotted,Mahogany] (-5.5,4.5) -- (-5,4);\n\t\t\\draw [dotted,Mahogany] (5,1) -- (5.5,1);\n\t\t\\draw [Mahogany](-5,3) to[out=-45,in=180] (-3.5,2) to[out=0,in=180] (-1.5,2) to[out=0,in=90] (-1,1.5) to[out=-90,in=0] (-1.5,1) to[out=180,in=0] (-3.5,1) to[out=180,in=-45] (-5,2);\n\t\t\\draw [Mahogany] (-5,4) to[out=-45,in=180] (-3.5,3) to[out=0,in=180] (-1.25,3) to[out=0,in=180] (1,1) -- (5,1);\n\t\t\\draw [Mahogany] (4,2.5) circle (.75cm);\n\t\t\\node at (-4.5,4) [Mahogany] {$V$};\n\t\t\n\t\t\\draw [|-|,very thick,gray] (-3.5,1) -- (-1.5,1);\n\t\t\\draw [|-|,very thick,gray] (-3.5,2) -- (-1.5,2);\n\t\t\\draw [|-|,very thick,gray] (-3.5,3) -- (-1.5,3);\n\t\t\\node at (-2.5,4) [gray,fill=white] {$V_0 \\times [-\\epsilon,\\epsilon]$};\n\t\t\n\t\t\\draw [|-|,very thick,Periwinkle] (1,1) -- (3,1);\n\t\t\\node at (2,4) [Periwinkle,fill=white] {$V_1 \\times [-\\epsilon,\\epsilon]$};\n\t\t\n\t\t\\node at (-2.8,3) {$\\bullet$};\n\t\t\\node at (-2.1,2) {$\\bullet$};\n\t\t\\node at (-1.8,2) {$\\bullet$};\n\t\t\\node at (4,1.75) {$\\bullet$};\n\t\t\\node at (.8,1) {$\\bullet$};\n\t\t\\node at (1.5,1) {$\\bullet$};\n\t\\end{tikzpicture}\n\t\\caption{An element of $\\mathrm{wall}_2^V$. We suppressed the framings at the points in the configuration indicated by the black points.}\n\t\\label{fig:wall-semi}\n\\end{figure}\n\nWe now supply the postponed ingredients to the proof of \\cref{prop:cutting-hocolim}. This finishes the proof of that proposition and thus also that of \\cref{prop:image-in-Morita}.\n\n\\begin{lem}\\label{lem:microfibration-lemma}\\ \n\\begin{enumerate}\n\t\\item \\label{enum:microfibration-lemma-i} The thick realisation of the map \\eqref{equ:resolve-embeddings-into-infty} is a weak equivalence.\n\t\\item\\label{enum:microfibration-lemma-ii} The map \\eqref{equ:bar-to-wall} is a levelwise weak equivalence.\n\t\\item\\label{enum:microfibration-lemma-iii} The augmentation $\\varepsilon\\colon \\mathrm{wall}_\\bullet^V\\rightarrow C_n^{\\mathrm{fr}}(V)$ realises to a weak equivalence.\n\t\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nWe begin with a general observation. Let $X$ be a nonempty totally ordered topological poset (by which we mean topological space $X$ with a total order on its underlying set). If the function $\\max(x_0,-)\\colon X\\rightarrow X$ is continuous for some $x_0\\in X$, then the nerve of $X$ is weakly contractible, since the sequence of inequalities $x\\le \\max(x_0,x)\\ge x_0$ induces a zig-zag of natural transformations from the identity on $X$ to the constant functor with values $x_0$, so we obtain a homotopy between the identity map of the nerve of $X$ and the constant map.\n\t\nReplacing the (half-)open intervals in the definition of $\\llparenthesis\\hspace{.1em} a \\hspace{.1em}\\rrparenthesis^\\ast$ with closed intervals, we get a weakly equivalent semisimplicial space. Doing so, by the parametrised isotopy extension theorem the augmentation \\eqref{equ:resolve-embeddings-into-infty} is a levelwise fibration. Hence to prove \\ref{enum:microfibration-lemma-i} it suffices to show that the semisimplicial space given by the fibres over an embedding $e\\colon \\llparenthesis\\hspace{.1em} a\\hspace{.1em}\\rrparenthesis^\\ast\\hookrightarrow \\llparenthesis\\hspace{.1em}\\infty\\hspace{.1em}\\rrparenthesis^\\ast=[-\\epsilon,\\epsilon]$ realises to a weakly contractible space. This agrees with the nerve of the nonempty totally ordered poset of real numbers $t\\in(-\\epsilon,\\epsilon)$ disjoint from the image of $e$, so the claim follows from the observation.\n\t\nTo show part \\ref{enum:microfibration-lemma-ii}, we choose for all $p\\ge0$ a function $\\tau\\colon [p]\\rightarrow(\\epsilon,\\epsilon)$ and an embedding $e\\in\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)$ such that $\\tau$ hits every component of the complement of $e$.  This induces \n\\vspace{-0.2cm}\n\\[\n\t(e\\circ(-),\\tau)\\colon \\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}}}(-,\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis)\\xlra{\\simeq} \\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(-,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)_p\n\\] \nwhich is an equivalence and induces the left vertical equivalence in the commutative diagram\n\\[\\begin{tikzcd}\n\tB\\big(C^\\mathrm{fr}_n,\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}},\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}}}(-,\\llparenthesis\\hspace{.1em} p \\hspace{.1em}\\rrparenthesis)\\big)\\rar{\\simeq}\\dar{\\simeq}&C_n^{\\mathrm{fr}}(V\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis)\\dar\\\\\n\tB\\big(C^\\mathrm{fr}_n,\\underline{\\ensuremath{\\cat{Cut}}_{\\mathrm{sur}}},\\mathrm{Map}_{\\underline{\\ensuremath{\\cat{Cut}}_\\mathrm{sur}^\\rhd}}(-,\\llparenthesis\\hspace{.1em} \\infty \\hspace{.1em}\\rrparenthesis)_p\\big)\\rar&\\mathrm{wall}_p^V\n\\end{tikzcd}\\]\nwhose top horizontal map is induced by composition and evaluation. The latter is a weak equivalence for the same reason as \\eqref{equ:bar-resolution}. The right vertical map is induced by the function $\\tau\\colon [p]\\rightarrow (-\\epsilon,\\epsilon)$ and the embedding $e$, and it is easily seen to be an equivalence as well, so the bottom horizontal map is an equivalence, as claimed.\n\t\nTo show that $\\|\\varepsilon\\|$ is a weak equivalence, note that its fibre at a framed configuration $\\vec{x}\\in C_n^\\mathrm{fr}(X)$ is the realisation of the nerve of the nonempty totally ordered topological poset of real numbers $t\\in (\\epsilon,\\epsilon)$ such that $\\{t\\}\\times V_i\\subset V$ is disjoint from $\\vec{x}$ for all $i=1,\\ldots k$, so it is weakly contractible by the above observation. We now show that $\\|\\varepsilon\\|$ is a microfibration, which will finish the proof because any microfibration with weakly contractible fibres is a weak equivalence by \\cite[Lemma 2]{WeissClassifying}. The remaining task is thus to show that given commutative solid arrows as in\n\\[\\begin{tikzcd}[ar symbol\/.style = {draw=none,\"\\textstyle#1\" description,sloped},\tsubset\/.style = {ar symbol={\\subset}},row sep=1cm,column sep=0.3cm]\n\t&D^i\\times{\\{0\\}} \\dar \\rar{f} &[5pt] {\\|\\mathrm{wall}_\\bullet^V\\|} \\dar{\\parallel\\varepsilon\\parallel}\\arrow[r,subset]&{\\|\\mathrm{wall}_\\bullet\\|\\times C^\\mathrm{fr}_n(V)} \\\\\n\tD^i \\times \\left[0,\\delta\\right]\\arrow[urr,dashed,crossing over, end anchor={south west},\"\\widetilde{\\psi}\" pos=0.3]\\arrow[r,subset] &D^i \\times [0,1] \\arrow[swap,r,\"\\psi\"] & C^\\mathrm{fr}_n(V)&\n\\end{tikzcd}\\]\nthere is an $0<\\delta\\le 1$ for which a dashed lift as indicated exists. For this, we note that the necessary data to lift a framed configuration $\\vec{x}\\in C^\\mathrm{fr}_n(V)$ to $\\|\\mathrm{wall}_\\bullet^V\\| \\subset \\|\\mathrm{wall}_\\bullet\\|\\times C^\\mathrm{fr}_n(V)$ is a point $z\\in\\mathrm{int}(\\Delta^p)$ for some number $p\\ge0$, a function $\\tau\\colon [p]\\rightarrow (-\\epsilon,\\epsilon)$ such that $\\vec{x}$ is disjoint from $\\{\\tau(i)\\}\\times V_j\\subset V$ for all $i$ and $j$. For any $\\vec{x}'$ close enough to $\\vec{x}$ the same data works, so for each $x\\in D^i$ we get lifts $\\psi(x,t)$ for $t\\in[0,\\delta_x]$ for some $0<\\delta_x\\le 1$, uses that the subspaces $V_i\\subset V$ are closed. By compactness, we can find a uniform choice of $\\delta_x$ for $x\\in D^i$. This gives the lift.\n\\end{proof}\n\n\\subsection{Unitality}\\label{step:functor-to-morita}The goal of this step is to prove the following proposition, which uses the terminology of \\cref{sec:quasi-unital} and its variation from \\cref{rem:quasi-unital-into-simplicial} \\ref{enum:quasi-unital-into-simplicial-i}.\n\n\\begin{prop}\\label{prop:quasi-unitality}The non-unital bordism category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\in\\ensuremath{\\mathrm{Cat}}_{\\mathrm{nu}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$ is quasi-unital and the following morphism of semisimplicial objects in $\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$ is quasi-unital\n\\[\n\tE^{\\mathrm{geo}}\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\longrightarrow\\ensuremath{\\mathrm{Fun}}_{\\Delta^\\mathrm{op}}(\\Delta^\\mathrm{op}_{\/[\\bullet]},\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup}).\n\\]\n\\end{prop}\n\nBy the equivalence \\eqref{equ:qu-is-good}, the non-unital double $\\infty$-category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}$ thus extends to a (unital) double $\\infty$-category \\[\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\in\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}).\\] The second part of the proposition together with \\cref{rem:quasi-unital-into-simplicial} \\ref{enum:quasi-unital-into-simplicial-ii} and \\cref{lem:image-in-premorita} implies that the composition \\eqref{equ:non-unital-composition} is quasi-unital in the sense of \\cref{rem:quasi-unital-into-simplicial} \\ref{enum:quasi-unital-into-simplicial-i}, so using the second part of this remark once more, together with \\cref{prop:image-in-Morita}, we conclude that the functor of double $\\infty$-categories $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\rightarrow \\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))$ is quasi-unital and thus extends by the equivalence \\eqref{equ:qu-is-good} essentially uniquely to a functor of double $\\infty$-categories\n\\[\n\tE\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d) \\longrightarrow \\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)).\n\\]\n\n\n\\begin{proof}[Proof of \\cref{prop:quasi-unitality}]\nThis is tedious but straight-forward, so we avoid spelling out all details.  Recalling that $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}$ is the levelwise coherent nerve of a semisimplicial $\\ensuremath{\\cat{Kan}}$-enriched category $\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}$, the quasi-unit is given by the coherent nerve of the simplicial functor $u\\colon \\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[0]}\\rightarrow \\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu})_{[1]}$ which sends a $[0]$-walled $d$-manifold $(W,\\mu)$ to $(\\ensuremath{\\mathbf{R}}\\times W|_{\\mu(0)},\\mu')$ with $\\mu'(0)=\\mu(0)$ and $\\mu'(1)=\\mu(0)+1$. On morphisms, it is induced by sending $\\varphi\\colon W|_{[\\mu(0)-\\epsilon,\\mu(0)+\\epsilon]}\\rightarrow W'|_{[\\mu'(0)-\\epsilon,\\mu'(0)+\\epsilon]}$ to $\\mathrm{id}_\\ensuremath{\\mathbf{R}}\\times \\varphi_{0}$. \n\nTo prove that the functor $E^{\\mathrm{geo}}\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\rightarrow\\ensuremath{\\mathrm{Fun}}_{\\Delta^\\mathrm{op}}({\\Delta^\\mathrm{op}}_{\/[\\bullet]},\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup})$ is quasi-unital, recall that it was constructed as the coherent nerve of the zig-zag\n\\vspace{-0.1cm}\n\\[\n\\ensuremath{\\cat{ncBord}}(d)^\\mathrm{nu}_{[\\bullet]} \\xra{E^{\\mathrm{geo}}_{[\\bullet]}}\\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} \\bullet\\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\cat{Man}}_d^\\sqcup) \t\\xla{\\simeq}\\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em}\\bullet\\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\cat{Man}}_d^\\sqcup) \\cong \\\n\t\\ensuremath{\\mathrm{Fun}}_{\\Delta^{\\mathrm{op}}}(\\Delta^{\\mathrm{op}}_{\/[\\bullet]},\\ensuremath{\\cat{Man}}_d^\\sqcup) \n\\]\nof semisimplicial objects in $\\ensuremath{\\cat{Kan}}$-enriched categories. We first construct the top horizontal functor in a commutative diagram of $\\ensuremath{\\cat{Kan}}$-enriched categories\n\\begin{equation}\\label{equ:0th-degeneracy}\n\\begin{tikzcd}\n\t\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} 1\\hspace{.1em}\\rrparenthesis\/}\\rar\\dar{\\simeq}& \\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} 0\\hspace{.1em}\\rrparenthesis\/}\\dar{\\simeq}\\\\[-2pt]\n\t\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} 1\\hspace{.1em}\\rrparenthesis\/}\\rar{\\iota^*}& \\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} 0\\hspace{.1em}\\rrparenthesis\/}\n\\end{tikzcd}\n\\end{equation}\nwhere  $\\iota\\colon \\llparenthesis\\hspace{.1em} 0\\hspace{.1em}\\rrparenthesis\\rightarrow\\llparenthesis\\hspace{.1em} 1\\hspace{.1em}\\rrparenthesis$ is the unique morphism. On objects, the top arrow agrees with the bottom one. On morphisms, the top arrow is given by sending an embedding $\\overline{\\gamma}\\colon \\mathrm{wlab}_{\\alpha}(\\ensuremath{\\mathbf{R}})|^{\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis}\\hookrightarrow \\mathrm{wlab}_{\\alpha'}(\\ensuremath{\\mathbf{R}})$ to the unique dashed embedding that makes the diagram\n\\[\\begin{tikzcd}[ar symbol\/.style = {draw=none,\"\\textstyle#1\" description,sloped},\tsubset\/.style = {ar symbol={\\supset}}]\n\t{\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis}\\times\\ensuremath{\\mathbf{R}}&[-15pt]\\arrow[l,subset]\\mathrm{wlab}_{\\alpha}(\\ensuremath{\\mathbf{R}})|^{\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis}\\arrow[d,hookrightarrow]\\arrow[r,two heads]&\\mathrm{wlab}_{\\alpha\\circ\\iota}(\\ensuremath{\\mathbf{R}})|^{\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis}\\arrow[d,hookrightarrow,dashed]\\arrow[r,equal]&[-15pt]{\\gamma^{-1}\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis}\\times(-\\epsilon,\\epsilon)\\\\\n\t\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}&\\arrow[l,subset]\\mathrm{wlab}_{\\alpha'}(\\ensuremath{\\mathbf{R}})\\arrow[r,two heads]&\\mathrm{wlab}_{\\alpha'\\circ\\iota}(\\ensuremath{\\mathbf{R}})\\arrow[r,equal] &\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis\\times(-\\epsilon,\\epsilon)\n\\end{tikzcd}\\]\ncommute where the bottom surjection is the identity if $\\alpha'(1)\\in\\{L,R\\}$ and otherwise the union of the identity over $\\llparenthesis\\hspace{.1em} \\mathring{q'}\\hspace{.1em}\\rrparenthesis\\backslash\\alpha'(1)$ with the map\n\\[\n\t\\mathrm{wlab}_{\\alpha'}(\\ensuremath{\\mathbf{R}})|^{\\alpha'(1)}=(-\\epsilon,\\epsilon]\\sqcup [1-\\epsilon,1+\\epsilon)\\xra{\\mathrm{tr}_{-\\epsilon}\\sqcup \\mathrm{tr}_{-(1-\\epsilon)}}(-2\\epsilon,2\\epsilon)\\xra{1\/4}(-\\epsilon,\\epsilon)=\\mathrm{wlab}_{\\alpha'\\circ\\iota}(\\ensuremath{\\mathbf{R}})|^{\\alpha'(1)}\n\\]\nover $\\alpha'(1)$; the top arrow is defined in the same way by replacing $\\alpha'$ by $\\alpha$. \nApplying $\\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(-,\\ensuremath{\\cat{Man}}_d^\\sqcup)$ to \\eqref{equ:0th-degeneracy} results in a commutative diagram of $\\ensuremath{\\cat{Kan}}$-enriched categories\n\\[\\begin{tikzcd}\n\t\\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} 0\\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\cat{Man}}_d^\\sqcup)\\rar\\dar{\\simeq}& \\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em} 1\\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\cat{Man}}_d^\\sqcup)\\dar{\\simeq}\\\\[-2pt]\n\t\\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} 0\\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\cat{Man}}_d^\\sqcup)\\rar& \\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\ensuremath{\\cat{Cut}}_{\\llparenthesis\\hspace{.1em} 1\\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\cat{Man}}_d^\\sqcup),\n\\end{tikzcd}\\]\nso $N_\\mathrm{coh}(-)$ applied to the top arrow models the $0$th degeneracy map of $\\ensuremath{\\mathrm{Fun}}_{\\Delta^{\\mathrm{op}}}(\\Delta^{\\mathrm{op}}_{\/[\\bullet]},\\ensuremath{\\icat{M}\\mathrm{an}}_d^\\sqcup)$. Using this model for the degeneracy and the above quasi-unit for $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}=N_{\\mathrm{coh}}(\\ensuremath{\\cat{ncBord}}(d)^{\\mathrm{nu}})$, it is tedious but straightforward to check that $N_\\mathrm{coh}(E^{\\mathrm{geo}}_\\bullet)$ and thus $E^{\\mathrm{geo}}$ is quasi-unital.\n\\end{proof}\n\n\n\\subsection{Symmetric monoidal structure}\\label{step:symmetric-monoidal-structure}In this step we promote the functor of double $\\infty$-categories $E\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d) \\rightarrow \\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))$ to a functor of \\emph{symmetric monoidal} double $\\infty$-categories (modelled as commutative monoid objects in $\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$, see \\cref{sec:monoidal-cats}). This is not difficult and essentially amounts to adding an index by a finite pointed set $\\langle s\\rangle\\in\\ensuremath{\\cat{Fin}}_*$ to the previous steps. To avoid being too repetitive, we will not spell out all details.\n\n\\begin{nconvention}Given a space $X$, a map $\\lambda\\colon X\\rightarrow \\langle s\\rangle$ to $\\langle s\\rangle$, and a subset $A\\subset \\langle s\\rangle$, we denote the preimage of $A$ by ${}^A|X\\coloneqq \\lambda^{-1}(A)$ to distinguish it from the notation $X|_A$ and $X|^A$ introduced in \\cref{conv:epsilon-conventions} and \\ref{step:mand}.\n\\end{nconvention}\n\n\\subsubsection*{\\ref{step:bordismcat}': the bordism category}\nWe first extend $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\in\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})$ to a symmetric monoidal non-unital double $\\infty$-category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\in \\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Cat}}_\\mathrm{nu}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))$ as follows: firstly, we extend the semisimplicial object $\\ensuremath{\\cat{ncBord}}(d)^{\\mathrm{nu}}\\in\\ensuremath{\\mathrm{Fun}}(\\Delta^\\mathrm{op}_\\mathrm{inj},\\ensuremath{\\cat{sCat}})$ in $\\ensuremath{\\cat{Kan}}$-enriched categories to an object $\\ensuremath{\\cat{ncBord}}(d)^{\\mathrm{nu}}\\in\\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{Fin}}_*,\\ensuremath{\\mathrm{Fun}}(\\Delta_\\mathrm{inj}^\\mathrm{op},\\ensuremath{\\cat{sCat}}))=\\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{Fin}}_*\\times \\Delta_\\mathrm{inj}^\\mathrm{op},\\ensuremath{\\cat{sCat}})$; evaluation at $\\langle 1\\rangle\\in\\ensuremath{\\cat{Fin}}_*$ recovers the previous construction. The value of $\\ensuremath{\\cat{ncBord}}(d)^{\\mathrm{nu}}$ at $([p],\\langle s\\rangle)$ for $\\langle s\\rangle\\in\\ensuremath{\\cat{Fin}}_*$ is the $\\ensuremath{\\cat{Kan}}$-enriched category ${\\ensuremath{\\cat{ncBord}}(d)^{\\mathrm{nu}}}_{[p],\\langle s\\rangle}$ whose objects are $[p]$-walled $d$-manifolds $(W,\\mu)$ together with a map $\\lambda\\colon W\\rightarrow \\langle\\mathring{s}\\rangle$, which we think of as a way to decompose $W$ into disjoint summands indexed by $\\langle\\mathring{s}\\rangle$. Morphisms from $(W,\\mu,\\lambda)$ to $(W',\\mu',\\lambda')$ are embeddings of $[p]$-walled manifolds that are additionally assumed to commute with the maps to $\\langle \\mathring{s}\\rangle$. The functoriality of ${\\ensuremath{\\cat{ncBord}}(d)^{\\mathrm{nu}}}_{[p],\\langle s\\rangle}$ in $p$ is defined as for ${\\ensuremath{\\cat{ncBord}}(d)^{\\mathrm{nu}}}_{[p]}$, and that in $\\langle s\\rangle$ is for $\\varphi\\in \\ensuremath{\\cat{Fin}}_*(\\langle s\\rangle, \\langle s'\\rangle)$ on objects given by $\\smash{(W,\\mu,\\lambda)\\mapsto ({}^{\\varphi^{-1}\\langle \\mathring{s}\\rangle}|W,\\mu,\\varphi \\circ \\lambda)}$ and on morphisms by restricting embeddings. A mild extension of the proof of \\cref{lem:bord-is-category-object} then shows that taking taking coherent nerves yields a commutative monoid object  in double $\\infty$-categories, as wished.\n\n\\subsubsection*{\\ref{step:mand}': the manifold category}\nNext, we extend the monoidal $\\infty$-category $\\ensuremath{\\icat{M}\\mathrm{an}}_d$ (thought of as a cocartesian fibration $\\ensuremath{\\icat{M}\\mathrm{an}}^{\\sqcup}_d\\rightarrow \\ensuremath{\\cat{Cut}}$) to an \\emph{symmetric} monoidal $\\infty$-category. It will be convenient to view it as a commutative monoid object in monoidal $\\infty$-categories $\\ensuremath{\\icat{M}\\mathrm{an}}_d\\in\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))\\subset \\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{Fin}}_*,\\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{Cut}},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))$. To this end, we extend the construction of the functor $\\ensuremath{\\cat{Man}}^{\\sqcup}_d\\rightarrow \\ensuremath{\\cat{Cut}}$ of $\\ensuremath{\\cat{Kan}}$-enriched categories to yield $\\ensuremath{\\cat{Kan}}$-enriched functors ${\\ensuremath{\\cat{Man}}_d}^{\\sqcup,\\langle s\\rangle}\\rightarrow \\ensuremath{\\cat{Cut}}$, one for each pointed set $\\langle s\\rangle\\in\\ensuremath{\\cat{Fin}}_*$. Objects of $\\smash{\\ensuremath{\\cat{Man}}^{\\sqcup,\\langle s\\rangle}_d}$ are now triples $(W,\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis,\\lambda)$ of $\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis\\in\\ensuremath{\\cat{Cut}}$, a smooth submanifold $W\\subset\\llparenthesis\\hspace{.1em}\\mathring{p}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^\\infty$ and a map $\\lambda\\colon W\\rightarrow \\langle\\mathring{s}\\rangle$. The space of morphisms is defined as before, with the additional requirement that the embeddings have to commute with the reference maps to $\\langle\\mathring{s}\\rangle$. Given a map $\\varphi\\colon \\langle s\\rangle\\rightarrow \\langle s'\\rangle$ in $\\ensuremath{\\cat{Fin}}_*$, there is a functor ${\\ensuremath{\\cat{Man}}_d}^{\\sqcup,\\langle s\\rangle}\\rightarrow {\\ensuremath{\\cat{Man}}_d}^{\\sqcup,\\langle s'\\rangle}$ over $\\ensuremath{\\cat{Cut}}$ which on objects is given by $(W,\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis,\\lambda)\\mapsto(^{\\varphi^{-1}\\langle \\mathring{s'}\\rangle}|W,\\llparenthesis\\hspace{.1em} p\\hspace{.1em}\\rrparenthesis,\\varphi\\circ \\lambda)$\n and on morphisms is induced by restriction. This yields a functor from $\\ensuremath{\\cat{Fin}}_*$ to cocartesian fibrations over $\\ensuremath{\\cat{Cut}}$. Using straightening and taking coherent nerves then gives the desired commutative monoid object in monoidal $\\infty$-categories.\n\n\\subsubsection*{\\ref{step:functor-to-premorita}': from the bordism category to the pre-Morita category of manifolds}\nBy the discussion in \\cref{sec:morita-functoriality}, taking pre-Morita categories of $\\ensuremath{\\icat{M}\\mathrm{an}}_d\\in\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))$ yields a commutative monoid object $\\smash{\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)\\in\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))}$, and our next task is to upgrade the morphism $E^{\\mathrm{geo}}\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\rightarrow \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)$ in $\\smash{\\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}}_\\mathrm{inj},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})}$ from \\ref{step:functor-to-premorita} to a morphism in $\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Fun}}(\\Delta_\\mathrm{inj}^{\\mathrm{op}},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))$. To do this, we first define for each $\\langle s\\rangle\\in\\ensuremath{\\cat{Fin}}_*$ a variant $\\smash{E^{\\mathrm{geo}}_{[\\bullet],\\langle s\\rangle}\\colon (\\ensuremath{\\cat{ncBord}}(d)^{\\mathrm{nu}})_{[\\bullet],\\langle s\\rangle}\\rightarrow \\ensuremath{\\mathrm{Fun}}_{\\ensuremath{\\cat{Cut}}}(\\underline{\\ensuremath{\\cat{Cut}}}_{\\llparenthesis\\hspace{.1em}\\bullet\\hspace{.1em}\\rrparenthesis\/},\\ensuremath{\\cat{Man}}_d^{\\sqcup,\\langle s\\rangle})}$ in $\\ensuremath{\\mathrm{Fun}}(\\Delta_\\mathrm{inj}^\\mathrm{op},\\ensuremath{\\cat{sCat}})$ of \\eqref{equ:psi-simplicially}. For this, note that in the notation of Substep \\ref{step:functor-to-premorita-language} I, projection on $\\ensuremath{\\mathbf{R}}\\times \\ensuremath{\\mathbf{R}}^\\infty$ gives a map $\\mathrm{lab}_\\alpha(W,\\mu)\\rightarrow W$ for any $[p]$-walled manifold, so if $W$ comes with a map to $\\langle\\mathring{s}\\rangle$, then so does $\\mathrm{lab}_\\alpha(W,\\mu)$. Based on this observation, the construction of $\\smash{E^{\\mathrm{geo}}_{[\\bullet]}}$ from \\eqref{equ:psi-simplicially} directly generalises to a functor $\\smash{E^{\\mathrm{geo}}_{[\\bullet],\\langle s\\rangle}}$ as desired by incorporating the maps to $\\langle s\\rangle$. Varying $s$, the maps $\\smash{E^{\\mathrm{geo}}_{[\\bullet],\\langle s\\rangle}}$ define a morphism in $\\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{Fin}}_*,\\ensuremath{\\mathrm{Fun}}(\\Delta_\\mathrm{inj}^{\\mathrm{op}},\\ensuremath{\\cat{sCat}}))$. Taking coherent nerves gives desired extension of $E^{\\mathrm{geo}}$ to a morphism in the full subcategory $\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Fun}}(\\Delta_\\mathrm{inj}^{\\mathrm{op}},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))\\subset \\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{Fin}}_*,\\ensuremath{\\mathrm{Fun}}(\\Delta_\\mathrm{inj}^{\\mathrm{op}},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))$,\n\\begin{equation}\\label{equ:psi-monoidal}\n\tE^{\\mathrm{geo}}\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\longrightarrow \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d).\n\\end{equation}\n\n\n\\subsubsection*{\\ref{step:composite}': composite algebras}\nWe claim that the two functor \\begin{equation}\\label{equ:extend-functor-monoidal}\\ensuremath{\\icat{M}\\mathrm{an}}_d\\longrightarrow \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)\\longrightarrow\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)\\end{equation} extend to  morphisms in $\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty})\\simeq \\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))$ (see \\cref{rem:comm-mon-iterative}). For the first map, we discussed this in \\cref{sec:presheaf-category}. A restriction map on presheaves such as the map only lax symmetric monoidal in general, but turns out to be actually monoidal in our case:\n\n\\begin{lem}\\label{lem:iota-monoidal} The lax symmetric monoidal functor $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d) \\to \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ induced by restriction along the inclusion $\\iota^* \\colon \\ensuremath{\\icat{D}\\mathrm{isc}}_d\\hookrightarrow \\ensuremath{\\icat{M}\\mathrm{an}}_d$ is strong monoidal.\n\\end{lem}\n\n\\begin{proof}By the formula for Day convolution it suffices to verify that for finite sets $S$ the inclusion $\\smash{(\\ensuremath{\\icat{D}\\mathrm{isc}}_d \\times \\ensuremath{\\icat{D}\\mathrm{isc}}_d)^\\mathrm{op}_{S \\times \\ensuremath{\\mathbf{R}}^d\/} \\subset (\\ensuremath{\\icat{M}\\mathrm{an}}_d \\times \\ensuremath{\\icat{M}\\mathrm{an}}_d)^\\mathrm{op}_{S \\times \\ensuremath{\\mathbf{R}}^d\/}}$ is cofinal (recall the convention to take slices before opposition). By \\cite[4.1.3.1]{LurieHTT} it suffices to prove that  $\\smash{((\\ensuremath{\\icat{D}\\mathrm{isc}}_d \\times \\ensuremath{\\icat{D}\\mathrm{isc}}_d)^\\mathrm{op}_{S \\times \\ensuremath{\\mathbf{R}}^d\/})_{\/u}}$ has a terminal object for all triples $(M,M',u)$ of $M,M' \\in \\ensuremath{\\icat{M}\\mathrm{an}}_d$ and $u \\colon S \\times \\ensuremath{\\mathbf{R}}^d \\hookrightarrow M \\sqcup M'$. Such a terminal object is given by the factorisation $\\smash{S \\times \\ensuremath{\\mathbf{R}}^d=T \\times \\ensuremath{\\mathbf{R}}^d\\sqcup T' \\times \\ensuremath{\\mathbf{R}}^d\\xrightarrow{u} M}$ where the decomposition $S=T\\sqcup T'$ is so that $T \\times \\ensuremath{\\mathbf{R}}^d = u^{-1}(M)$ and $T' \\times \\ensuremath{\\mathbf{R}}^d = u^{-1}(M')$.\n\\end{proof}\n\nAfter applying $\\overline{\\ensuremath{\\mathrm{ALG}}}(-)$ to \\eqref{equ:extend-functor-monoidal} this gives a composition of morphisms in $\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}}_\\mathrm{inj},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))$ (see \\cref{sec:morita-functoriality}) which we may precompose with \\eqref{equ:psi-monoidal} to arrive at an enhancement of \\eqref{equ:non-unital-composition} to \\begin{equation}\\label{equ:non-unital-composition-monoidal} \n\t\\overline{E} \\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}}\\xlra{E^{\\mathrm{geo}}}\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)\\xlra{y_*}\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d))\\xlra{\\iota^*}\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))\n\\end{equation}\nin $\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}}_\\mathrm{inj},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))\\subset \\ensuremath{\\mathrm{Fun}}(\\ensuremath{\\cat{Fin}}_*\\times \\Delta^{\\mathrm{op}}_\\mathrm{inj},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))$. To show that this composition lands in the levelwise full subcategory $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))\\subset \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))$ (which lies in the full subcategory $\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))\\subset \\ensuremath{\\mathrm{CMon}}(\\smash{\\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}}_\\mathrm{inj}},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))$, see \\cref{sec:morita-functoriality}), by the Segal property it suffices to show this after evaluation at $\\langle 1\\rangle \\in\\ensuremath{\\cat{Fin}}_*$ where it agrees with the previously variant without symmetric monoidal structures for which we have already checked this property in \\ref{step:composite}, so we obtain a map $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\mathrm{nu}} \\rightarrow \\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))$ in $\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Cat}}_{\\mathrm{nu}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))$. Finally, a minor extension of the arguments of \\ref{step:functor-to-morita} to incorporate indexing maps to finite sets enhances this to a functor of symmetric monoidal double $\\infty$-categories\t$E\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\rightarrow  \\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))$.\n\n\\subsection{Variants}\\label{step:variants}\nWe now define several variants of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$, related by a diagram \n\\begin{equation}\\label{equ:variants-of-ncbord}\n\\begin{tikzcd}[row sep=0.3cm]\n\t\\gls*{borddcat}\\dar\\rar& \\gls*{borddcatbdy}\\dar\\rar&\\ensuremath{\\icat{B}\\mathrm{ord}}(d-1)\\dar \\\\\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\rar &\\gls*{borddcatbdync}\\rar&\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d-1)\n \\end{tikzcd}\n\\end{equation}\nof symmetric monoidal double $\\infty$-categories. Informally speaking, $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$ is obtained from $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ by restricting to compact bordisms between closed manifolds, the versions with a $(-)^{\\partial}$-subscript allow manifolds to have boundary, all vertical maps and the left horizontal maps are induced by inclusion, and the right horizontal maps are induced by taking boundaries.\n\n\\subsubsection{Compact variant}\nTo define the compact variant $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$, we say that a $[p]$-walled $d$-manifold $(W,\\mu)$ is of \\emph{of compact type} if the subspace $W|_{[\\mu(0)-\\epsilon,\\mu(p)+\\epsilon]} \\subseteq W$ is compact. Restricting to $[p]$-walled $d$-manifolds of compact type in the construction of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ defines the symmetric monoidal double $\\infty$-category $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$. By construction, it comes with a levelwise full subcategory inclusion into $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$. This is the leftmost vertical map in \\eqref{equ:variants-of-ncbord}.\n\n\\subsubsection{Variants with boundary}\nTo define the variant $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\partial}$ of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ involving manifolds with boundary, we replace $[p]$-walled $d$-manifolds $(W,\\mu)$ where $W\\subset \\ensuremath{\\mathbf{R}}\\times \\ensuremath{\\mathbf{R}}^{\\infty}$ is required to have no boundary, by \\emph{$[p]$-walled $d$-manifolds with boundary}: these are pairs $(W,\\mu)$ of a smooth submanifold $W\\subset \\ensuremath{\\mathbf{R}}\\times[0,\\infty)\\times\\ensuremath{\\mathbf{R}}^{\\infty}$, possibly with boundary, together with an order-perserving function $\\mu\\colon [p]\\rightarrow\\ensuremath{\\mathbf{R}}$ such that\n\\begin{enumerate}\n\t\\item $(W,\\mu)$ satisfies the conditions in the definition of $[p]$-walled $d$-manifolds (see \\ref{step:bordismcat}),\n\t\\item\\label{enum:boundary-condition} $\\partial W=W\\cap (\\ensuremath{\\mathbf{R}}\\times\\{0\\}\\times\\ensuremath{\\mathbf{R}}^{\\infty})$ such that $W\\cap (\\ensuremath{\\mathbf{R}}\\times[0,\\epsilon]\\times\\ensuremath{\\mathbf{R}}^{\\infty})=\\partial W\\times [0,\\epsilon]$ under the appropriate identifications.\n\\end{enumerate}\nThe space $\\ensuremath{\\mathrm{Emb}}((W,\\mu),(W',\\mu'))$ of embeddings of $[p]$-walled $d$-manifolds with boundary is defined in the same way as in the case without boundary, except that we demand in addition that the embedding $\\varphi\\colon W|_{[\\mu(0)-\\epsilon,\\mu(p)+\\epsilon]}\\hookrightarrow W'|_{[\\mu'(0)-\\epsilon,\\mu'(p)+\\epsilon]}$ also satisfies\n\\begin{enumerate}\n\t\\item $\\varphi^{-1}(\\ensuremath{\\mathbf{R}}\\times [0,\\epsilon]\\times\\ensuremath{\\mathbf{R}}^{\\infty})=(W|_{[\\mu(0)-\\epsilon,\\mu(p)+\\epsilon]})\\cap (\\ensuremath{\\mathbf{R}}\\times [0,\\epsilon]\\times\\ensuremath{\\mathbf{R}}^{\\infty})$,\n\t\\item\\label{enum:emb-collared} under the appropriate identifications, $\\varphi$ restricts to an embedding of the form \n\t\\[\n\t\t(\\partial \\phi\\times \\mathrm{id}_{[0,\\epsilon]})\\colon \\partial W|_{[\\mu(0)-\\epsilon,\\mu(p)+\\epsilon]}\\times [0,\\epsilon]\\lhook\\joinrel\\longrightarrow  \\partial W'|_{[\\mu'(0)-\\epsilon,\\mu'(p)+\\epsilon]}\\times [0,\\epsilon]\n\t\\] \n\tfor some embedding $\\partial \\varphi\\colon \\partial W|_{[\\mu(0)-\\epsilon,\\mu(p)+\\epsilon]}\\hookrightarrow \\partial W'|_{[\\mu(0)-\\epsilon,\\mu(p)+\\epsilon]}$.\n\\end{enumerate}\nReplacing the $[p]$-walled $d$-manifolds in the construction of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ by $[p]$-walled $d$-manifolds with boundaries in the sense just described gives rise to a symmetric monoidal double $\\infty$-category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\partial}$ which receives a levelwise full subcategory inclusion from $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$, induced by the inclusion $\\ensuremath{\\mathbf{R}}\\times\\ensuremath{\\mathbf{R}}^{\\infty} \\cong \\ensuremath{\\mathbf{R}}\\times\\{1\\}\\times \\ensuremath{\\mathbf{R}}^{\\infty}\\subset \\ensuremath{\\mathbf{R}}\\times[0,\\infty)\\times \\ensuremath{\\mathbf{R}}^{\\infty}$. This inclusion restricts to a functor $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)\\rightarrow \\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{\\partial}$ where $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$ is the symmetric monoidal double $\\infty$-category given as the levelwise full subcategory of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\partial}$ obtained by restricting to $[p]$-walled $d$-manifolds with boundary of compact type, defined by the same condition as for the variant without boundary. This explains \\eqref{equ:variants-of-ncbord}, except for the horizontal functor of the right square which are induced by sending a $[p]$-walled $d$-manifold with boundary $W\\subset \\ensuremath{\\mathbf{R}}\\times[0,\\infty)\\times \\ensuremath{\\mathbf{R}}^\\infty$ to its boundary $\\partial W=W\\cap(\\ensuremath{\\mathbf{R}}\\times\\{0\\}\\times \\ensuremath{\\mathbf{R}}^\\infty)$, with the same walls, and restricting embeddings to the boundary. \n\n\\subsubsection{Tangential structures without boundary}\\label{sec:tangential-no-bdy}\nAssociating to a smooth manifold $M$ its frame bundle $\\mathrm{Fr}(M)$ with its canonical right $\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}})$-action, induces a functor of $\\ensuremath{\\cat{Kan}}$-enriched categories \n\\begin{equation}\\label{equ:simplicial-frame-bundle}\n\t(\\ensuremath{\\cat{Man}}_d^{\\sqcup})_{[1]}\\longrightarrow \\ensuremath{\\mathrm{Fun}}(\\mathrm{GL}^{\\mathrm{op}}_d,\\cat{S})^{\\circ},\n\\end{equation}\nwhere $\\ensuremath{\\cat{Man}}_d^{\\sqcup}$ is the symmetric monoidal $\\infty$-category from \\ref{step:mand} and \\ref{step:symmetric-monoidal-structure}, and $\\mathrm{GL}_d$ is the (singular simplicial set of) the topological group $\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}})$ viewed as a $\\ensuremath{\\cat{Kan}}$-enriched groupoid with one object. The superscript $(-)^\\circ$ indicates that we pass to the full subcategory on the fibrant-cofibrant objects in the projective model structure on $\\ensuremath{\\mathrm{Fun}}(\\mathrm{GL}^\\mathrm{op}_d,\\cat{S})$, as in \\cite[A.3.3.2]{LurieHA}. Let us explain why functor $\\mathrm{Fr}(-)$ takes values in this subcategory. Firstly $\\mathrm{Fr}(M)$ is fibrant: in the projective model structure an object is fibrant if its underlying simplicial set is a Kan complex, and this is the case for $\\mathrm{Fr}(M)$ as a singular simplicial set of a topological space. Secondly $\\mathrm{Fr}(M)$ is cofibrant: because the map $\\ensuremath{\\mathrm{Fun}}(\\mathrm{GL}_d^\\mathrm{op},\\cat{S}) \\to \\cat{S}$ that forgets the action is the right adjoint in a Quillen adjunction, each map $\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}}) \\times S \\to \\mathrm{GL}_d(\\ensuremath{\\mathbf{R}}) \\times S'$ with canonical right $\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}})$-action and $S \\to S'$ a monomorphism is a cofibration, and $\\mathrm{Fr}(M)$---being locally trivial---is isomorphic to a (possibly transfinite) composition of pushouts against such maps.\n\nApplying coherent nerves to the map \\eqref{equ:simplicial-frame-bundle} and viewing $\\mathrm{GL}_d$ as an $\\infty$-category via the coherent nerves, gives a functor of $\\infty$-categories\n\\[\n\t(\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup})_{[1]}\\simeq N_{\\mathrm{coh}}((\\ensuremath{\\cat{Man}}_d^{\\sqcup})_{[1]})\\rightarrow N_{\\mathrm{coh}}(\\ensuremath{\\mathrm{Fun}}(\\mathrm{GL}^{\\mathrm{op}}_d,\\cat{S})^{\\circ})\\simeq \\ensuremath{\\mathrm{Fun}}(N_{\\mathrm{coh}}(\\mathrm{GL}^{\\mathrm{op}}_d),\\ensuremath{\\catsingle{S}})=\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)\n\\]\nwhere the second equivalence is an instance of \\cite[4.2.4.4]{LurieHTT}. Since the unit $\\varnothing\\in (\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\sqcup})_{[1]}$ is initial and so $\\ensuremath{\\icat{M}\\mathrm{an}}_d^\\sqcup$ is unital as an $\\infty$-operad  \\cite[2.3.1.1]{LurieHA}, this functor extends uniquely to a lax symmetric monoidal functor $\\mathrm{Fr}(-)\\colon \\ensuremath{\\icat{M}\\mathrm{an}}^{\\sqcup}\\rightarrow \\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)^\\sqcup$ where $\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)$ carries the cocartesian symmetric monoidal structure \\cite[2.4.3.9]{LurieHA}. Note that $\\mathrm{Fr}(M)\\sqcup \\mathrm{Fr}(N)\\rightarrow \\mathrm{Fr}(M\\sqcup N)$ is an equivalence for manifolds $M$ and $N$, so this is actually (strong) symmetric monoidal.\n\nBy an easier version of the argument in \\ref{step:composite}, the composition\n\\[\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\xlra{E^{\\mathrm{geo}}}\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)\\xrightarrow{\\overline{\\ensuremath{\\mathrm{ALG}}}(\\mathrm{Fr}(-))}\\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d))\n\\]\nlands in the Morita double $\\infty$-category $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d))\\subset \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d))$, which is equivalent to $\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d))$ (see \\cref{sec:alg-cospans}). We thus arrive at a functor of symmetric monoidal double $\\infty$-categories $\\mathrm{Fr}(-)\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\rightarrow\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d))$.\nInformally, this is given by sending a bordism $W\\colon P \\leadsto Q$ to the cospan $\\mathrm{Fr}(c(P))\\rightarrow \\mathrm{Fr}(W)\\leftarrow\\mathrm{Fr}(c(Q))$, where $c(P),c(Q)\\subset W$ are collar neighbourhoods of the boundary components. \n\n\\begin{dfn}\\label{dfn:bord-tang-structures} Given a \\emph{tangential structure} $\\gls*{theta}\\in \\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)$, we define $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d)$ and $\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(d)$ by the following pullbacks in symmetric monoidal double $\\infty$-categories\n\\[\\begin{tikzcd} \n\t\\gls*{borddcattheta} \\dar\\rar& \\gls*{ncborddcattheta} \\rar \\dar &  \\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)_{\/\\theta}) \\dar\\\\\n\t\\ensuremath{\\icat{B}\\mathrm{ord}}(d)\\rar{\\subset}&\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d) \\rar{\\mathrm{Fr}(-)} & \\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d));\n\\end{tikzcd}\\]\nhere the rightmost vertical map is induced by the forgetful functor $\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)_{\/\\theta}\\rightarrow \\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)$ which preserves colimits \\cite[1.2.13.8]{LurieHA} and thus induces a functor on cospan categories.\n\\end{dfn}\n\nVarying $\\theta$ induces functors $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^{(-)}(d),\\ensuremath{\\icat{B}\\mathrm{ord}}^{(-)}(d)\\colon  \\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)\\rightarrow \\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Cat}}(\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))$. In particular, for a map $\\theta\\rightarrow\\theta'$ in $\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)$, we have functors\n\\begin{equation}\\label{eqn:bord-theta-naturality}\n\t\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(d) \\longrightarrow \\ensuremath{\\icat{B}\\mathrm{ord}}^{\\theta'}(d)\\quad\\text{and}\\quad\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d) \\longrightarrow \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^{\\theta'}(d)\n\\end{equation}\n\n\\subsubsection{Tangential structures with boundary}\nTo define the version $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^\\partial$ that includes tangential structure, one uses a variant \n\\begin{equation}\\label{equ:egeo-with-boundary}\n\tE^{\\mathrm{geo}}\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^\\partial\\longrightarrow \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}^\\partial_d).\n\\end{equation} \nof the map $E^{\\mathrm{geo}}\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\rightarrow \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)$ between commutative monoid objects in simplicial $\\infty$-categories. The symmetric monoidal $\\infty$-category $\\ensuremath{\\icat{M}\\mathrm{an}}^\\partial_d$ is defined in the same way as $\\ensuremath{\\icat{M}\\mathrm{an}}_d$ except that we use submanifolds $W\\subset \\llparenthesis\\hspace{.1em} \\mathring{p}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\\times[0,\\infty)\\times\\ensuremath{\\mathbf{R}}^\\infty$ that may have boundary, but have to satisfy the evident analogue of \\ref{enum:boundary-condition} in the definition of a $[p]$-walled $d$-manifold with boundary. With this modification, the construction in \\ref{step:functor-to-premorita} and its extensions in \\ref{step:functor-to-morita} and \\ref{step:symmetric-monoidal-structure} extends almost verbatim to give the map \\eqref{equ:egeo-with-boundary} in $\\smash{\\ensuremath{\\mathrm{CMon}}(\\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}},\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}))}$.\n\nAssigning to a manifold $W\\in (\\ensuremath{\\icat{M}\\mathrm{an}}^{\\partial,\\sqcup}_d)_{[1]}$ the map $\\mathrm{Fr}(\\partial W\\times[0,\\epsilon])\\rightarrow \\mathrm{Fr}(W)$ induced by the inclusion induces an extension of the functor $(\\ensuremath{\\icat{M}\\mathrm{an}}_d^\\sqcup)_{[1]}\\rightarrow\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)$ to a functor of $\\infty$-categories \n\\[\n\t(\\ensuremath{\\icat{M}\\mathrm{an}}^{\\partial,\\sqcup}_d)_{[1]}\\longrightarrow \\ensuremath{\\mathrm{Fun}}([1]\\times N_{\\mathrm{coh}}(\\mathrm{GL}_d^\\mathrm{op}),\\ensuremath{\\catsingle{S}})\\eqcolon\\ensuremath{\\mathrm{PSh}}([1]\\times \\mathrm{GL}_d)\n\\] \nwhich, by the same argument as in the case without boundary, extends to a symmetric monoidal functor $\\mathrm{Fr}(-)\\colon \\ensuremath{\\icat{M}\\mathrm{an}}^{\\partial,\\sqcup}_d\\rightarrow \\ensuremath{\\mathrm{PSh}}([1]\\times \\mathrm{GL}_d)^\\sqcup$ where the target is equipped with the cocartesian symmetric monoidal structure. This functor allows us to extend \\cref{dfn:bord-tang-structures} to define symmetric monoidal double $\\infty$-categories \n\\[\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^{\\theta}(d)^\\partial \\quad\\text{ and }\\quad\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^{\\theta}(d)^\\partial\n\\]\nfor any \\emph{tangential structure $\\theta$ with boundary} by which mean a map $\\theta=(\\theta^\\partial{\\rightarrow}\\theta^\\circ)\\in \\ensuremath{\\mathrm{PSh}}([1]\\times \\mathrm{GL}_d)$.\n \n\\subsubsection{Taking boundaries with tangential structures}\\label{sec:boudnary-functor-tangential-structure}\nNext, we extend the ``taking-boundaries functors'' $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\partial}\\rightarrow \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d-1)$ and $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{\\partial}\\rightarrow \\ensuremath{\\icat{B}\\mathrm{ord}}(d-1)$ from \\eqref{equ:variants-of-ncbord} to include tangential structures. This involves the commutative diagram of $\\infty$-categories\n\\begin{equation}\\label{equ:res-ind-tang-structures}\n\\begin{tikzcd}\n\t(\\ensuremath{\\icat{M}\\mathrm{an}}_d^{\\partial,\\sqcup})_{[1]}\\dar\\rar&\\ensuremath{\\mathrm{PSh}}([1]\\times \\mathrm{GL}_d)\\arrow[r,equal]&\\ensuremath{\\mathrm{PSh}}([1]\\times \\mathrm{GL}_d)\\dar{\\mathrm{res}}\\\\\n\t(\\ensuremath{\\icat{M}\\mathrm{an}}^\\sqcup_{d-1})_{[1]}\\rar&\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_{d-1})\\rar{\\mathrm{ind}_{d-1}^d}&\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_{d})\n\\end{tikzcd}\n\\end{equation}\nwhere the leftmost vertical map is induced by sending a submanifold $W\\subset \\llparenthesis\\hspace{.1em} \\mathring{p}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\\times[0,\\infty)\\times\\ensuremath{\\mathbf{R}}^\\infty$ to its boundary, i.e.\\,the intersection with $\\llparenthesis\\hspace{.1em}\\mathring{p}\\hspace{.1em}\\rrparenthesis\\times\\ensuremath{\\mathbf{R}}\\times\\{0\\}\\times\\ensuremath{\\mathbf{R}}^\\infty$. The arrow labelled $\\mathrm{res}$ is induced by precomposition with the inclusion $\\{1\\}\\times\\mathrm{GL}_d\\subset [1]\\times\\mathrm{GL}_d$ and arrow labelled $\\mathrm{ind}_{d-1}^d$ is the left adjoint to the functor $\\mathrm{res}_{d-1}^d\\colon \\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_{d})\\rightarrow \\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_{d-1})$ induced by precomposition with the inclusion $\\mathrm{GL}_{d-1}(\\ensuremath{\\mathbf{R}})\\subset\\mathrm{GL}_{d}(\\ensuremath{\\mathbf{R}})$ using the first $(d-1)$-coordinates. One way to provide the commutativity of \\eqref{equ:res-ind-tang-structures} is to recognise this diagram as the coherent nerve of a diagram of $\\ensuremath{\\cat{Kan}}$-enriched categories (using \\cite[5.2.4.6]{LurieHTT} for $\\mathrm{ind}_{d-1}^d$) and then use the fact that the extension $\\mathrm{ind}_{d-1}^d(\\mathrm{Fr}(\\partial W))\\rightarrow \\mathrm{Fr}(\\partial W\\times[0,\\epsilon])$ of the $\\mathrm{GL}_{d-1}(\\ensuremath{\\mathbf{R}})$-equivariant map $\\mathrm{Fr}(\\partial W)\\rightarrow\\mathrm{Fr}(\\partial W\\times[0,\\epsilon])$ induced by the inclusion $\\partial W\\times\\{0\\}\\subset \\partial W\\times[0,\\epsilon]$ and the canonical non-zero vector field on $[0,\\epsilon]$ is an equivalence of $\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}})$-spaces which is natural in $W$.\n\nEquipping all categories of presheaves with the cocartesian symmetric monoidal structure and using the universality property as in \\ref{sec:tangential-no-bdy}, we can extend \\eqref{equ:res-ind-tang-structures} to a commutative diagram of symmetric monoidal $\\infty$-categories. Applying $\\overline{\\ensuremath{\\mathrm{ALG}}}(-)$, using the $E^\\mathrm{geo}$-functors, and the equivalence $\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\catsingle{C}})\\simeq\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\catsingle{C}})$ for cocartesian $\\ensuremath{\\catsingle{C}}$, this leads to a commutative diagram of symmetric monoidal double $\\infty$-categories\n\\[\n\t\\begin{tikzcd}[column sep=0.3cm]\n\t\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^\\partial\\rar\\dar&\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^\\partial\\dar\\rar&\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}([1]\\times \\mathrm{GL}_d))\\arrow[r,equal]&\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}([1]\\times \\mathrm{GL}_d))\\dar{(\\mathrm{res})_*}\\\\\n\t\\ensuremath{\\icat{B}\\mathrm{ord}}(d-1)^\\partial\\rar&\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d-1)\\rar&\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_{d-1}))\\rar{(\\mathrm{ind}_{d-1}^d)_*}&\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_{d}))\n\t\\end{tikzcd}\n\\]\nFor a tangential structure with boundary $\\theta=(\\theta^\\partial{\\rightarrow}\\theta^\\circ)\\in \\ensuremath{\\mathrm{PSh}}([1]\\times \\mathrm{GL}_d)$, this induces extensions\n\\[\n\t\\ensuremath{\\icat{B}\\mathrm{ord}}^{\\theta}(d)^\\partial\\longrightarrow \\ensuremath{\\icat{B}\\mathrm{ord}}^{\\mathrm{res}_{d-1}^d(\\theta^\\partial )}(d-1)\\quad\\text{and}\\quad \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^{\\theta}(d)^\\partial\\longrightarrow \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^{\\mathrm{res}_{d-1}^d(\\theta^\\partial )}(d-1)\n\\]\nof the ``taking boundaries'' functors from \\eqref{equ:variants-of-ncbord}.\n\n\\begin{ex}\\label{ex:framing-to-oneframing}\nThe tangential structure with boundary encoding framings is $\\mathrm{fr}\\coloneq (\\mathrm{id}\\colon \\mathrm{GL}_d(\\ensuremath{\\mathbf{R}})\\rightarrow\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}}))$, so the above in particular gives a functor of symmetric monoidal double $\\infty$-categories\n$\\ensuremath{\\icat{B}\\mathrm{ord}}^\\mathrm{fr}(d)^{\\partial}\\rightarrow \\ensuremath{\\icat{B}\\mathrm{ord}}^{\\mathrm{1{-}fr}}(d-1)^{\\partial}$\nfrom the compact framed $d$-dimensional bordism category with boundary to the $d$-dimensional bordism category with boundary and the tangential structure $\\mathrm{1{-}fr}\\coloneq \\mathrm{res}_{d-1}^d(\\mathrm{GL}_{d}(\\ensuremath{\\mathbf{R}}))$ encodes framings of the once-stablised tangent bundle. \n\\end{ex}\n\n\\subsection{Product functors}\\label{step:product}\nGiven a smooth $p$-manifold $P$, possibly with boundary, we now explain the construction of a ``taking products'' functor of symmetric monoidal double $\\infty$-categories\n\\begin{equation}\\label{equ:product-functor-noncompact-bord}\n\t(P\\times-)\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^\\partial\\longrightarrow \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d+p)^\\partial,\n\\end{equation}\nwhich restricts to product functors of the form \n\\[\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\rightarrow \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d+p),\\quad\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^\\partial\\rightarrow \\ensuremath{\\icat{B}\\mathrm{ord}}(d+p)^\\partial,\\quad \\text{and} \\quad\\ensuremath{\\icat{B}\\mathrm{ord}}(d)\\rightarrow \\ensuremath{\\icat{B}\\mathrm{ord}}(d+p)\n\\]\nif $P$ has no boundary, is compact, or is closed, respectively. This will involve smoothing corners.\n\nWe fix an embedding $P\\subset [0,\\infty)\\times\\ensuremath{\\mathbf{R}}^N$ for some $N\\ge0$ which satisfies the condition \\ref{enum:boundary-condition} in the definition of a $[p]$-walled $d$-manifolds with boundary (ignoring the first $\\ensuremath{\\mathbf{R}}$-factor). Furthermore, we fix once and for all a homeomorphism $\\psi\\colon [0,\\infty)\\times[0,\\infty)\\rightarrow[0,\\infty)\\times\\ensuremath{\\mathbf{R}}$ such that\n\\begin{enumerate}\n\t\\item \\label{enum:psi-boundary} $\\psi$ agrees with the identity on $[0,\\infty)\\times \\{0\\}$ and with the counterclockwise rotation by $\\pi\/2$ on $\\{0\\}\\times [0,\\infty)$. In particular, it fixes the origin.\n\t\\item \\label{enum:psi-diff} $\\psi$ is a diffeomorphism away from the origin.\n\t\\item \\label{enum:psi-collar} $\\psi^{-1}([0,\\epsilon]\\times\\ensuremath{\\mathbf{R}})\\subset \\big([0,\\epsilon]\\times[0,\\infty)\\cup [0,\\infty)\\times [0,\\epsilon]\\big)$,\n\t\\item \\label{enum:psi-delta} $\\psi([0,\\delta]\\times[0,\\infty)\\cup [0,\\infty)\\times [0,\\delta])\\subset [0,\\epsilon]\\times \\ensuremath{\\mathbf{R}}$ for some fixed $0<\\delta\\le \\epsilon$\n\t\\item \\label{enum:psi-1-1} $\\psi$ fixes the point $(1,1)$.\n\\end{enumerate}\nUsing $\\psi$ and its properties \\ref{enum:psi-boundary}--\\ref{enum:psi-collar}, given a $[p]$-walled $d$-manifold with boundary $(W,\\mu)$, we obtain a $[p]$-walled $(d+p)$-manifold with boundary $(\\Psi(P\\times W),\\mu)$ with $\\Psi(P\\times W)$ the image of $P\\times W$ under the composition\n\\[\n\t[0,\\infty)\\times\\ensuremath{\\mathbf{R}}^N\\times \\ensuremath{\\mathbf{R}}\\times[0,\\infty)\\times\\ensuremath{\\mathbf{R}}^\\infty\\xrightarrow{\\text{swap}}\\ensuremath{\\mathbf{R}}\\times [0,\\infty)\\times [0,\\infty)\\times\\ensuremath{\\mathbf{R}}^N\\times \\ensuremath{\\mathbf{R}}^\\infty\\xrightarrow{\\mathrm{id}_\\ensuremath{\\mathbf{R}}\\times \\psi\\times \\text{shift}}\\ensuremath{\\mathbf{R}}\\times [0,\\infty)\\times\\ensuremath{\\mathbf{R}}^\\infty\n\\]\nwhere the first map swaps the $\\ensuremath{\\mathbf{R}}^N$-factor with the middle $(\\ensuremath{\\mathbf{R}}\\times[0,\\infty))$-factor. Note this comes with a preferred homeomorphism $P\\times W\\cong\\Psi(P\\times W)$ which is a diffeomorphism away from $\\partial P\\times\\partial W$. Taking products with $P$ and conjugating with $\\psi$ induces a map \n\\[\n\t\\ensuremath{\\mathrm{Emb}}\\big((W,\\mu),(W',\\mu')\\big)\\longrightarrow \\ensuremath{\\mathrm{Emb}}\\big((\\Psi(P\\times W),\\mu),(\\Psi(P\\times W'),\\mu')\\big),\n\\]\nwhich is well-defined due to the collaring condition \\ref{enum:emb-collared} on embeddings between $[p]$-walled $d$-manifolds with boundaries. Going through the construction of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^\\partial$, one checks that the assignment $(W,\\mu)\\mapsto (\\Psi(P\\times W),\\mu)$ together with the maps between embedding spaces just discussed leads to functors as desired.\n\nThese product functors can be extended to include tangential structures. To this end, one notes that there is a functor of symmetric monoidal categories \\[(P\\times-)\\colon \\ensuremath{\\icat{M}\\mathrm{an}}_d^\\partial\\longrightarrow\\ensuremath{\\icat{M}\\mathrm{an}}_{p+d}^\\partial\\] defined as for \\eqref{equ:product-functor-noncompact-bord}. On underlying $\\infty$-categories, this participates in a diagram of $\\infty$-categories\n\\begin{equation}\\label{equ:product-with-tang-structures}\n\\begin{tikzcd}\n\t(\\ensuremath{\\icat{M}\\mathrm{an}}_d^\\partial)_{[1]}\\arrow[dd,\"{(P\\times-)}\"]\\rar&\\ensuremath{\\mathrm{PSh}}([1]\\times\\mathrm{GL}_d)\\dar\\\\[-10pt]\n\t&\\ensuremath{\\mathrm{PSh}}([1]\\times\\mathrm{GL}_p\\times\\mathrm{GL}_d)\\dar{\\mathrm{ind}_{p,d}^{p+d}}\\\\\n\t(\\ensuremath{\\icat{M}\\mathrm{an}}_{p+d}^\\partial)_{[1]}\\rar&\\ensuremath{\\mathrm{PSh}}([1]\\times\\mathrm{GL}_{p+d})\n\\end{tikzcd}\n\\end{equation}\nwhere the upper right vertical arrow is the functor that sends a map $X\\rightarrow Y$ of $\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}})$-spaces to \n\\[(\\mathrm{Fr}(P)\\times X)\\cup_{\\mathrm{Fr}(\\partial P\\times [0,\\epsilon])\\times X}(\\mathrm{Fr}(\\partial P\\times [0,\\epsilon])\\times Y)\\rightarrow \\mathrm{Fr}(P)\\times Y\\] viewed as a map of $(\\mathrm{GL}_{p}(\\ensuremath{\\mathbf{R}})\\times\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}}))$-spaces, and the functor $\\mathrm{ind}_{p,d}^{p+d}$ is the left adjoint to the restriction along the inclusion $\\mathrm{GL}_p(\\ensuremath{\\mathbf{R}})\\times\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}})\\subset \\mathrm{GL}_{p+d}(\\ensuremath{\\mathbf{R}})$. \\eqref{equ:product-with-tang-structures} can be extended to a \\emph{commutative} square of $\\infty$-categories in a way similar to what we did for \\eqref{equ:res-ind-tang-structures}: recognise it as the coherent nerve of a diagram of $\\ensuremath{\\cat{Kan}}$-enriched categories and then use that the two compositions are related by a zig-zag of natural equivalences. In this case, the zig-zag is provided by the commutative diagram\n\\[\\hspace{-.3cm} \\begin{tikzcd}[column sep=-0.2cm, row sep=0.3cm]\n\t\\mathrm{res}^{p+d}_{p,d}\\mathrm{Fr}(\\partial \\Psi(P\\times W)\\times[0,\\epsilon])\\rar&\\mathrm{res}^{p+d}_{p,d}\\mathrm{Fr}(\\Psi(P\\times W))\\\\\n\t(\\mathrm{Fr}(\\mathrm{int}(P))\\times \\mathrm{Fr}(c(W))))\\cup_{\\mathrm{Fr}(c(P))\\times \\mathrm{Fr}(c(W))}(\\mathrm{Fr}(c(P))\\times \\mathrm{Fr}(\\mathrm{int}(W))\\uar\\rar\\dar&\\mathrm{Fr}(\\mathrm{int}(P))\\times\\mathrm{Fr}(\\mathrm{int}(W))\\uar\\dar\\\\\n\t(\\mathrm{Fr}(P)\\times \\mathrm{Fr}(\\partial W\\times[0,\\epsilon]))\\cup_{\\mathrm{Fr}(\\partial P\\times [0,\\epsilon]))\\times \\mathrm{Fr}(W\\times [0,\\epsilon])}(\\mathrm{Fr}(\\partial P\\times[0,\\epsilon])\\times \\mathrm{Fr}(W)\\rar&\\mathrm{Fr}(P)\\times\\mathrm{Fr}(W)\n\\end{tikzcd}\\]\nof $(\\mathrm{GL}_{p}(\\ensuremath{\\mathbf{R}})\\times\\mathrm{GL}_{d}(\\ensuremath{\\mathbf{R}}))$-spaces which is natural in $W$ and consists of vertical equivalences when taking adjoints with respect to the $(\\smash{\\mathrm{ind}_{p,d}^{p+d}},\\smash{\\mathrm{res}_{p,d}^{p+d}})$-adjunction. Here $c(P)\\coloneq \\partial P\\times(0,\\delta)\\subset \\mathrm{int}(P)$ and $c(W)\\coloneq \\partial W\\times(0,\\delta)\\subset \\mathrm{int}(P)$, the lower vertical arrows are induced by the inclusions $\\mathrm{int}(P)\\subset P$ and $\\mathrm{int}(W)\\subset W$, and the upper vertical arrows by the preferred embedding $\\mathrm{int}(P)\\times\\mathrm{int}(W)\\hookrightarrow \\Psi(P\\times W)$ induced by $\\psi$; this uses property \\ref{enum:psi-delta} of $\\psi$. Similarly to the final paragraph of \\ref{sec:boudnary-functor-tangential-structure}, \\eqref{equ:product-with-tang-structures} yields a commutative diagram of symmetric monoidal double $\\infty$-categories\n\\begin{equation}\\label{equ:products-with-tang-structure-bord}\n\\begin{tikzcd}\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^{\\partial}\\arrow[d,\"P\\times(-)\",swap]\\rar&\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}([1]\\times\\mathrm{GL}_d))\\dar\\\\\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(p+d)^{\\partial}\\rar&\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}([1]\\times\\mathrm{GL}_{p+d})).\n\\end{tikzcd}\n\\end{equation}\nNow given a tangential structure with boundary $\\lambda=(\\lambda^\\partial\\rightarrow \\lambda^\\circ)\\in \\ensuremath{\\mathrm{PSh}}([1]\\times\\mathrm{GL}_p)$, and a $\\lambda$-structure on $P$ in the form of a map $\\ell_P\\colon (\\mathrm{Fr}(P),\\mathrm{Fr}(\\partial P\\times [0,\\epsilon]))\\rightarrow (\\lambda^\\circ,\\lambda^{\\partial})$ in $\\ensuremath{\\mathrm{PSh}}([1]\\times\\mathrm{GL}_p)$, then \\eqref{equ:products-with-tang-structure-bord} induces a functor of symmetric monoidal double $\\infty$-categories\n\\[\n\t((P,\\ell_P)\\times(-))\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^{\\theta}(d)^\\partial\\longrightarrow\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^{\\mathrm{glue}(\\theta,\\lambda)}(p+d)^\\partial\n\\]\nwhere $\\mathrm{glue}(\\theta,\\lambda)\\coloneq \\mathrm{ind}_{p,d}^{p+d}\\big(\\lambda^\\circ\\times\\theta^\\partial\\cup_{\\lambda^\\partial\\times\\theta^\\partial}\\lambda^\\partial\\times\\theta^\\circ \\rightarrow \\lambda^\\circ\\times\\theta^\\circ\\big)\\in \\ensuremath{\\mathrm{PSh}}([1]\\times\\mathrm{GL}_{p+d})$. This also extends the variants of the product functors mentioned below \\eqref{equ:product-functor-noncompact-bord}, where property \\ref{enum:psi-1-1} of $\\psi$ is used for the variants without boundary.\n\n\\begin{ex}\\label{ex:product-functor-with-framing}\nIn the case of framings $\\mathrm{fr}_p=\\lambda=(\\mathrm{id}\\colon \\mathrm{GL}_p(\\ensuremath{\\mathbf{R}})\\rightarrow \\mathrm{GL}_p(\\ensuremath{\\mathbf{R}}))$ and $\\mathrm{fr}_d=\\theta=(\\mathrm{id}\\colon \\mathrm{GL}_d(\\ensuremath{\\mathbf{R}})\\rightarrow \\mathrm{GL}_d(\\ensuremath{\\mathbf{R}}))$, we have $\\mathrm{glue}(\\mathrm{fr}_p,\\mathrm{fr}_d)\\simeq \\mathrm{fr}_{d+p}$, so omitting the subscripts, we have a product functor of symmetric monoidal double $\\infty$-categories $((P,\\ell_P)\\times(-))\\colon\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\mathrm{fr}(d)^\\partial\\rightarrow\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\mathrm{fr}(p+d)^\\partial$\nfor framed $p$-manifolds $P$, and similarly for the compact variants.\n\\end{ex}\n \n\\renewcommand\\thesubsection{\\thesection.\\arabic{subsection}}\n\n\\section{Properties of $E$, embedding calculus, and $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces}\\label{sec:functor-e-disc-structure}  \nThe main outcome of the previous section is the construction of a functor \n\\[\n\tE\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\longrightarrow \\gls*{modd} \\coloneqq \\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))\n\\]\nof symmetric monoidal double $\\infty$-categories, in the sense of \\cref{sec:monoidal-cats}, from a bordism category of (possible noncompact) $(d-1)$-manifolds to a Morita category on the category $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ of presheaves on a category $\\ensuremath{\\icat{D}\\mathrm{isc}}_d$ of finite disjoint unions of $d$-dimensional Euclidean spaces and codimension $0$ embeddings between them. We also constructed variants $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$, $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^\\partial$, and $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^\\partial$ of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$, related by a diagram of symmetric monoidal double $\\infty$-categories \\eqref{equ:variants-of-ncbord}, as well as enhancements with tangential structures of all of these bordism categories.\n\nThis section has several purposes: firstly in \\cref{sec:bordism-mapping-oo-cats}, we give more practical descriptions of these double $\\infty$-categories by describing their objects and mapping $\\infty$-categories in a model-independent and more intuitive manner, and we explain the functor $E$ in these terms. For most of the arguments in the later sections, this discussion is sufficient and there is no need to know the specifics of the construction in \\cref{sec:the-functor}. Secondly, we establish three properties of the functor $E$: \n\\begin{itemize}\n\t\\item a descent property in \\cref{sec:descent}, \n\t\\item a close relationship to Goodwillie--Weiss' embedding calculus in \\cref{sec:embedding-calculus}, and \n\t\\item an isotopy extension property in \\cref{sec:isotopy-extension}.\n\\end{itemize} \nFinally, in \\cref{sec:disc-structure-spaces}, we give the definition of the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces we will work with.\n\n\\subsection{Mapping $\\infty$-categories}\\label{sec:bordism-mapping-oo-cats}\nRecall from \\cref{sec:mapping-infinity-category} that a double $\\infty$-category $\\ensuremath{\\catsingle{C}}$ has mapping $\\infty$-categories $\\ensuremath{\\catsingle{C}}_{A,B}$ for objects $A,B\\in\\ensuremath{\\catsingle{C}}_{[0]}$, and these feature in composition functors $\\ensuremath{\\catsingle{C}}_{A,B}\\times\\ensuremath{\\catsingle{C}}_{B,C}\\rightarrow\\ensuremath{\\catsingle{C}}_{A,C}$. We now spell these out for some of the double $\\infty$-categories of the previous section.\n\n\\subsubsection{$\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$} \\label{sec:details-ncbord} \nIn short: objects of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ are (possibly noncompact) $(d-1)$-manifolds $P$ without boundary, and given two such manifolds $P$ and $Q$, the objects of the mapping $\\infty$-category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$ are bordisms $W\\colon P\\leadsto Q$ and the mapping spaces in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$ are given by embedding spaces relative to the boundary. The composition in these mapping $\\infty$-categories is by composing embeddings, the composition functor\n$\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q} \\times \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{Q,R} \\rightarrow \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,R}$ by gluing bordisms, and the symmetric monoidal structure by disjoint union.\n\nMore precisely, given a $(d-1)$-manifold $P$, we may use the weak Whitney embedding theorem to choose an embedding $P\\subset\\ensuremath{\\mathbf{R}}^{\\infty}$ and can thus view $P$ as a $[0]$-walled manifold $(\\ensuremath{\\mathbf{R}}\\times P,\\mu)$ in the sense of \\ref{step:bordismcat} of \\cref{sec:the-functor} (and hence as an object in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{[0]}$) by setting $\\mu(0)=0$. Moreover, it is easy to see that each object in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{[0]}$ is equivalent to one of this form, so we will no longer distinguish between abstract $(d-1)$-manifolds and objects in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{[0]}$. Similarly, given a bordism $W\\coloneq P\\leadsto Q$ between $(d-1)$-manifolds, we may embed it suitably collared in $[0,1]\\times \\ensuremath{\\mathbf{R}}^\\infty$ so that $((\\infty,0]\\times P\\cup W\\cup[1,\\infty)\\times Q,\\mu)$ with $\\mu(i)=i$ for $i=0,1$ is a $[1]$-walled manifold and thus an object in the mapping $\\infty$-category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$. Again, any object is equivalent to one of this form, so we will also no longer distinguish between abstract bordisms $P\\leadsto Q$ and objects in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$. The identification of the mapping spaces in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$ is justified by:\n\n\\begin{lem}\\label{lem:mapping-space-emb}\nGiven possibly noncompact bordisms $W,W'\\colon P\\leadsto Q$ between $(d-1)$-manifolds $P,Q$ without boundary, there is a natural equivalence $\t\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}}(W,W')\\simeq\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')$ in $\\ensuremath{\\catsingle{S}}$.\n\\end{lem}\n\n\\begin{proof}\nUsing that mapping spaces in a pullback of $\\infty$-categories are pullbacks of the mapping spaces, and that coherent nerves of $\\ensuremath{\\cat{Kan}}$-enriched categories preserve mapping spaces, we see $\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}}(W,W')$ is the fibre (i.e.\\,pullback along the indicated inclusion of a point) in $\\ensuremath{\\catsingle{S}}$ \n\\vspace{-0.2cm}\n\\[\n\t\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}}(W,W')=\\mathrm{fib}_{\\mathrm{id}}\\big(\\ensuremath{\\mathrm{Emb}}(W,W')_w\\xlra{\\mathrm{res}} \\ensuremath{\\mathrm{Emb}}(P,P)\\times \\ensuremath{\\mathrm{Emb}}(Q,Q) \\big)\n\\]\nwhere the subscript $(-)_w$ indicates that we restrict to the subspace of embeddings $e$ that in some fixed collars $P \\times [0,1] \\hookrightarrow W$ and $Q \\times [-1,0] \\hookrightarrow W$ have the form $\\mathrm{id}\\times e_P$ and $\\mathrm{id}\\times e_Q$ for self-embeddings $e_P$ and $e_Q$ of $P$ and $Q$ respectively. The map $\\mathrm{res}$ is induced by restriction to $e_P$ and $e_Q$. It is not hard to see that this is a Kan fibration, so the fibre in $\\ensuremath{\\catsingle{S}}$ agrees with the point-set fibre over $(\\mathrm{id},\\mathrm{id})$. The latter is $\\ensuremath{\\mathrm{Emb}}_{\\partial}(W,W')$, so we obtain an equivalence as claimed.\n\\end{proof} \n\n\\subsubsection{$\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$}\\label{sec:details-bord}\nUnder the identification of the objects in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{[0]}$ as $(d-1)$-manifolds $P$ without boundary, those in the full subcategory $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{[0]}$ correspond to $(d-1)$-manifolds $P$ without boundary that are also compact. Similarly, the objects in the mapping $\\infty$-categories of the levelwise full subcategory $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)\\subset \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ correspond to \\emph{compact} bordisms between closed manifolds. Since $\\ensuremath{\\mathrm{Emb}}_{\\partial}(W,W')=\\ensuremath{\\mathrm{Diff}}_\\partial(W,W')$ for two compact manifolds $W,W'$ with identified boundary, the morphism spaces in the mapping $\\infty$-categories are given by spaces of diffeomorphisms fixing the boundary. In particular, unlike for $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$, all mapping $\\infty$-categories of $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$ are $\\infty$-groupoids and can thus be regarded as spaces. Thus, by the discussion of \\cref{sec:double-vs-infty2}, not much is lost by applying $(-)^{(\\infty,1)}$ and instead consider the symmetric monoidal $\\infty$-category $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}$ with closed $(d-1)$-manifolds as objects and mapping spaces\n\\[\n\t\\textstyle{\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P,Q}\\simeq\\mathrm{Map}_{\\ensuremath{\\icat{B}\\mathrm{ord}}^{(\\infty,1)}(d)}(P,Q) \\simeq \\bigsqcup_{[W]} \\ensuremath{\\mathrm{BDiff}}_\\partial(W)},\n\\]\nwhere $[W]$ ranges over compact bordisms $W \\colon P \\leadsto Q$ up to diffeomorphism relative to the ends. Composition is by gluing bordisms and the symmetric monoidal structure by disjoint union.\n\n\\subsubsection{$\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d)$}\\label{sec:details-tangential-bord} \nIn short: given a tangential structure $\\theta$ in the form of a $\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}})$-space $\\theta$, the objects of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d)$ are (possibly noncompact) $(d-1)$-manifolds $P$ with a $\\theta$-structure on their once-stabilised tangent bundle, i.e.\\,a $\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}})$-equivariant map $\\theta_P\\colon \\mathrm{Fr}(I\\times N) \\rightarrow \\theta$ where $\\mathrm{Fr}(-)$ denotes the frame bundle and $I=[0,1]$. The objects of the mapping category $\\smash{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta}(d)_{(P,\\theta_P),(Q,\\theta_Q)}$ are bordisms with $\\theta$-structures and the morphisms are $\\theta$-embeddings, fixed on the boundary. The composition and monoidal structure is as in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$, but with the addition of $\\theta$-structures.\n\nTo justify this, recall from \\cref{sec:tangential-no-bdy} that the noncompact bordism category with $\\theta$-structures is defined as the pullback of symmetric monoidal double $\\infty$-categories\n\\[\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d) =\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d) \\times_{\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d))} \\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)_{\/\\theta}),\n\\]\nso the claimed description of the objects follows by using that forgetting symmetric monoidal structures preserves pullbacks and that pullbacks of double $\\infty$-categories are computed levelwise. This also shows that the mapping $\\infty$-categories are given by pullbacks of $\\infty$-categories\n\\[\\begin{tikzcd}\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d)_{(P,\\theta_P),(Q,\\theta_Q)} \\rar \\dar & \\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)_{\/\\theta})_{\\theta_P,\\theta_Q} \\dar \\\\\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q} \\rar & \\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d))_{\\mathrm{Fr}(I \\times P),\\mathrm{Fr}(I \\times Q)}\n\\end{tikzcd}\\]\nwhich justifies the description of the objects in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d)_{(P,\\theta_P),(Q,\\theta_Q)}$ when combined with the equivalence $\\ensuremath{\\mathrm{COSPAN}}^+(\\ensuremath{\\catsingle{C}})_{A,B}\\simeq \\ensuremath{\\catsingle{C}}_{A\\sqcup B\/}$ mentioned in \\cref{sec:span-cospan-cats}. Combining this discussion with the fact that mapping spaces in a pullback of $\\infty$-categories agree with the pullback of the mapping spaces, we arrive at the following precise version of the description of the mapping spaces in the mapping $\\infty$-category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d)_{(P,\\theta_P),(Q,\\theta_Q)}$.\n\n\\begin{lem}\\label{lem:mapping-space-theta}\nGiven $(d-1)$-manifolds $(P,\\theta_P),(Q,\\theta_Q)$ without boundary together with $\\theta$-structures on their once-stabilised tangent bundle, and $\\theta$-bordisms $(W,\\theta_W),(W',\\theta_{W'})\\colon (P,\\theta_P)\\leadsto (Q,\\theta_Q)$, there is a natural pullback diagram in $\\ensuremath{\\catsingle{S}}$.\n\\[\\hspace{-.25cm}\\begin{tikzcd}\n\t\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d)_{(P,\\theta_P),(Q,\\theta_Q)}}((W,\\theta_W),(W',\\theta_{W'})) \\dar \\rar &[-15pt] \\mathrm{Map}_{(\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)_{\/\\theta})_{\\theta_P \\sqcup \\theta_Q\/}}(\\theta_W,\\theta_{W'}) \\dar\\\\\n\t\\ensuremath{\\mathrm{Emb}}_\\partial(W,W') \\rar & \\mathrm{Map}_{\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)_{\\mathrm{Fr}(I \\times P)\\sqcup \\mathrm{Fr}(I \\times Q)\/}}(\\mathrm{Fr}(W),\\mathrm{Fr}(W'))).\n\\end{tikzcd}\\]\n\\end{lem}\n\n\\subsubsection{$\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(d)$}\\label{sec:details-tangential-bord-compact}\nThe previous discussion of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d)$  applies also to the levelwise full subcategory $\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(d)$ when restricting to compact manifolds throughout. By a minor enhancement of the argument in \\cref{sec:details-bord}, the mapping $\\infty$-categories $\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(d)$ are again $\\infty$-groupoids, so as for $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$ not much is lost by applying $(-)^{(\\infty,1)}$ and consider the symmetric monoidal $\\infty$-category $\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(d)^{(\\infty,1)}$ with closed $(d-1)$-manifolds with $\\theta$-structure on their once-stabilised tangent bundle as objects and mapping spaces given by \n\\begin{equation}\\label{equ:mapping-cat-theta-bord-compact}\n\t\\textstyle{\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(d)_{(P,\\theta_P),(Q,\\theta_Q)}\\simeq \\mathrm{Map}_{\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(d)^{(\\infty,1)}}((P,\\theta_P),(Q,\\theta_Q))\\simeq \\bigsqcup_{[W]} \\ensuremath{\\mathrm{BDiff}}^\\theta_\\partial(W,\\theta_P \\sqcup \\theta_Q)}\n\\end{equation}\nwhere $[W]$ ranges over compact bordisms $W \\colon P \\leadsto Q$ up to diffeomorphism relative to the ends and $\\ensuremath{\\mathrm{BDiff}}^\\theta_\\partial(W,\\theta_P \\sqcup \\theta_Q)$ is the quotient $\\mathrm{Map}_{\\ensuremath{\\mathrm{PSh}}(\\mathrm{GL}_d)_{\\mathrm{Fr}(I \\times P) \\sqcup \\mathrm{Fr}(I \\sqcup Q)\/}}(\\mathrm{Fr}(W),\\theta) \/ \\ensuremath{\\mathrm{Diff}}_\\partial(W)$ where the action is induced by precomposition (by standard bundle theory, this agrees with other definitions of $\\ensuremath{\\mathrm{BDiff}}^\\theta_\\partial(-)$ in the literature such as that in \\cite[Definition 1.5]{GRWstable}). Composition is given by gluing $\\theta$-bordisms and the symmetric monoidal structure by disjoint union.\n\n\\subsubsection{Variants with boundary}The discussion for the variants $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)^\\partial$ and $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^\\partial$ with boundary and their enhancements with tangential structures $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d)^\\partial$ and $\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(d)^\\partial$ is the same as that for the version without boundary, except that we allow the $(d-1)$-manifolds that appear as objects to have boundary and the bordisms $W\\colon P\\leadsto Q$ to be bordisms of manifolds with boundary. The bordisms thus come with a decomposition $\\partial W= \\partial_0W \\cup \\partial^hW\\cup\\partial_1W$ into codimension $0$ submanifolds where the \\emph{ends} $\\partial_iW$s are disjoint and come with identifications $P\\cong \\partial_0W$ and $Q\\cong  \\partial_1W$, and the \\emph{horizontal boundary} $\\partial^hW$ meets the ends in a corner. Embeddings between such manifolds are required to preserve this decomposition, map the interior to the interior, and be the identity near the ends, but they are allowed to move the horizontal boundary. With this convention, embeddings between \\emph{compact} bordisms are again diffeomorphisms. The discussion for the variants $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}^\\theta(d)^\\partial$ and $\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(d)^\\partial$ with tangential structures is similar; on the ends the tangential structures are fixed, but not on the horizontal boundary.\n\n\\subsubsection{$\\ensuremath{\\icat{D}\\mathrm{isc}}_d$ and $\\ensuremath{\\icat{M}\\mathrm{od}}(d)$}\\label{sec:details-bimod} \nThe objects of the symmetric monoidal $\\infty$-category $\\ensuremath{\\icat{D}\\mathrm{isc}}_d$ can be identified with $d$-manifolds without boundary that are diffeomorphic to a finite disjoint union of $\\ensuremath{\\mathbf{R}}^d$'s. The mapping spaces are given by codimension $0$ embeddings and the symmetric monoidal structure by disjoint union. Day convolution equips the $\\infty$-category $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ of $\\ensuremath{\\catsingle{S}}$-valued presheaves with a symmetric monoidal structure, and the objects of $\\ensuremath{\\icat{M}\\mathrm{od}}(d)=\\ensuremath{\\mathrm{ALG}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))$ are associative algebras in $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ (see the discussion in \\cref{sec:haugseng-morita}). The mapping $\\infty$-category between two associative algebras $A,B \\in \\ensuremath{\\icat{M}\\mathrm{od}}(d)$ is the $\\infty$-category $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{A,B}$ of $(A,B)$-bimodules and bimodule maps between these (see \\cref{sec:assalg-bimodules} where this category is denoted $\\ensuremath{\\mathrm{Bimod}}_{A,B}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$). The composition functors $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{A,B} \\times \\ensuremath{\\icat{M}\\mathrm{od}}(d)_{B,C} \\rightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)_{A,C}$ are given taking tensor products over $B$, which we denote by $(-)\\cup_{B}(-)$ to emphasise the similarity with the bordism category. The symmetric monoidal structure is given by external tensor product.  \n\n\n\\subsubsection{The functor $E$}\\label{sec:functor-e-summary}\nIn terms of the identifications of the objects and mapping categories of source and target explained in Sections~\\ref{sec:details-ncbord} and \\ref{sec:details-bimod}, the functor $E \\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d) \\rightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)$ of symmetric monoidal double $\\infty$-categories is on objects given by sending a $(d-1)$-manifold $P$ to the presheaf $E_{P\\times I}=\\ensuremath{\\mathrm{Emb}}(-,P\\times I)$ where $I=[0,1]$, equipped with the algebra structure induced by ``stacking''. On mapping $\\infty$-categories, it is given by the functor $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q} \\rightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}$ which sends a bordism $W \\colon M \\leadsto N$ to the presheaf $E_W=\\ensuremath{\\mathrm{Emb}}(-,W)$ with its $(E_{P \\times I},E_{Q \\times I})$-bimodule structure by ``stacking'', using fixed collars $P\\times I\\hookrightarrow W$ and $Q\\times I\\hookrightarrow W$ of both ends, where the convention is that the canonical vector field on $P \\times I$ is inwards pointing and that of $Q\\times I$ is outwards pointing. On morphisms, it sends an embedding $W \\hookrightarrow W'$ that is fixed on the boundary to the map $E_W\\rightarrow E_{W'}$ induced by postcomposition. That $E$ is a functor of double $\\infty$-categories in particular says that, given bordisms $W\\colon P\\leadsto Q$ and $W'\\colon Q\\leadsto R$, we have a preferred equivalence $E_{W\\cup_QW'}\\simeq E_W\\cup_{E_{Q\\times I}}E_{W'}$ of $(E_{P\\times I},E_{R\\times I})$-bimodules.\n\nWe will often restrict the functor $E$ to the levelwise full subcategory $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$ of $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ and pass to underlying symmetric monoidal $\\infty$-categories (i.e.\\,apply the functor $(-)^{(\\infty,1)}$ from \\cref{sec:double-vs-infty2}, which has little effect on $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$; see \\cref{sec:details-bord}) to obtain a functor of symmetric monoidal $\\infty$-categories $E\\colon \\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}\\rightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1)}$ . Recall from \\cref{sec:double-vs-infty2} that the mapping spaces of $\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1)}$ are given as $\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1)}}(A,B)\\simeq\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{A,B}$.\n\n\\subsection{Descent with respect to Weiss $\\infty$-covers}\\label{sec:descent}\nWe now prove a descent property for the mapping spaces in $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P\\times I},E_{Q\\times I}}$ for (possibly noncompact) $(d-1)$-manifolds $P$ and $Q$ without boundary. To state it, given a bordism $W\\colon P\\leadsto Q$, we write $\\ensuremath{\\catsingle{O}}(W)$ for the poset of open subsets of $W$ containing a neighbourhood of the boundary, ordered by inclusion. A subposet $\\ensuremath{\\catsingle{U}}\\subset\\ensuremath{\\catsingle{O}}(M)$ is a \\emph{Weiss $\\infty$-cover} of $M$ if any finite subset of $M$ is contained in some $O\\in\\ensuremath{\\catsingle{U}}$. Such a cover is \\emph{complete} if it contains a Weiss $\\infty$-cover for $\\bigcap_{O\\in \\ensuremath{\\catsingle{U}}'} O$ for any finite subset $\\ensuremath{\\catsingle{U}}'\\subset\\ensuremath{\\catsingle{U}}$. A functor $F\\colon\\ensuremath{\\catsingle{O}}(W)\\rightarrow \\ensuremath{\\catsingle{C}}$ to an $\\infty$-category $\\ensuremath{\\catsingle{C}}$ satisfies \\emph{descent for Weiss $\\infty$-covers} if for every $O\\in\\ensuremath{\\catsingle{O}}(W)$ and every complete Weiss $\\infty$-cover $\\ensuremath{\\catsingle{U}}\\subset \\ensuremath{\\catsingle{O}}(O)$ the diagram $F(O)\\rightarrow\\{F(U)\\}_{U\\in \\ensuremath{\\catsingle{U}}}$ is a limit diagram.\n\n\\begin{prop}\\label{prop:descent}For a bordism $W\\in \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$ and a bimodule $X\\in\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}$, the functor $\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}}(E_{(-)},X)\\colon\\ensuremath{\\catsingle{O}}(W)\\rightarrow \\ensuremath{\\catsingle{S}}$ satisfies descent for Weiss $\\infty$-covers.\n\\end{prop}\n\n\\begin{proof}\nIt suffices to show that for a given complete Weiss $\\infty$-cover $\\ensuremath{\\catsingle{U}}\\subset\\ensuremath{\\catsingle{O}}(O)$ of $O\\in\\ensuremath{\\catsingle{O}}(W)$, the diagram $\\{E_U\\}_{U\\in\\ensuremath{\\catsingle{U}}}\\rightarrow E_O$ is a colimit diagram in $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}=\\ensuremath{\\mathrm{Bimod}}_{E_{P \\times I},E_{Q \\times I}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))$. Since $\\ensuremath{\\catsingle{U}}$ is cofiltered, its nerve is weakly contractible so by \\cref{lemma:free-modules} \\ref{enum:free-modules-ii}, it suffices to show that the diagram is a colimit diagram after applying the forgetful functor to $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$. The result is is the diagram $\\{\\ensuremath{\\mathrm{Emb}}(-,U)\\}_{U\\in \\ensuremath{\\catsingle{U}}}\\rightarrow \\ensuremath{\\mathrm{Emb}}(-,O)$ in $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$, so as colimits in functor categories are computed objectwise \\cite[5.1.2.3]{LurieHTT}, it suffices to show  that $\\{\\ensuremath{\\mathrm{Emb}}(T\\times \\ensuremath{\\mathbf{R}}^d,U)\\}_{U\\in \\ensuremath{\\catsingle{U}}}\\rightarrow \\ensuremath{\\mathrm{Emb}}(T\\times \\ensuremath{\\mathbf{R}}^d ,O)$ is a colimit diagram in $\\ensuremath{\\catsingle{S}}$ for all finite sets $T$, or equivalently, that it is a homotopy colimit diagram in the Kan--Quillen model structure on simplicial sets. This holds by a well-known argument; see the proof of \\cite[Lemma 6.7]{KnudsenKupers}.\n\\end{proof}\n\n\\subsection{Relationship to embedding calculus}  \\label{sec:embedding-calculus} Using \\cref{prop:descent}, we now relate the functor $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}_{P,Q}\\rightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)_{P\\times I,Q\\times I}$ induced by $E$ on mapping $\\infty$-categories to the map $\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')\\rightarrow T_\\infty\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')$ provided by \\emph{embedding calculus} as introduced in \\cite{WeissImmersion,WeissImmersionErrata}.\n\n\\begin{thm}\\label{thm:emb-calc}\nGiven bordisms $W,W'\\in\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$, the map\n\\begin{equation}\\label{equ:emb-calc}\n\t\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}}(W,W')\\longrightarrow \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}}(E_W,E_{W'})\n\\end{equation}\nagrees up to equivalence with the map $\\ensuremath{\\mathrm{Emb}}_\\partial(W,W') \\to T_\\infty\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')$ from \\cite{WeissImmersion}.\n\\end{thm}\n\n\\begin{proof}\nWe consider the poset $\\ensuremath{\\catsingle{U}}$ of open subsets $U\\subset W$ that are unions $U=c(P) \\cup D\\cup c(Q)$ of three disjoint open subsets of $W$ where $c(P)$ and $c(Q)$ are open collars of the boundary components $P$ and $Q$ and $D$ is diffeomorphic to $T\\times\\ensuremath{\\mathbf{R}}^d$ for some finite set $T$, ordered by inclusion. Considering $U\\in\\ensuremath{\\catsingle{U}}$ as an object in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$, we obtain a commutative square in $\\ensuremath{\\catsingle{S}}$\n\\[\n\t\\begin{tikzcd}\n\t\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}}(W,W')\\dar{\\circled{1}}\\rar&\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}}(E_W,E_{W'}))\\dar{\\circled{3}}\\\\\n\t\\lim_{U\\in\\ensuremath{\\catsingle{U}}} \\big(\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}}(U,W')\\big)\\rar{\\circled{2}}& \\lim_{U\\in\\ensuremath{\\catsingle{U}}} \\big(\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}}(E_{U},E_{W'})\\big).\n\t\\end{tikzcd}\n\\]\nwhose vertical arrows are induced by restriction. By \\cref{lem:mapping-space-emb} the map $\\circled{1}$ agrees with the restriction map $\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')\\rightarrow \\lim_{U\\in\\ensuremath{\\catsingle{U}}}\\ensuremath{\\mathrm{Emb}}_\\partial(U,W')$ which in turn agrees with the map $\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')\\rightarrow T_\\infty\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')$ by the discussion in \\cite[Sections 5, 10]{WeissImmersion}, so the claim follows once we show that $\\circled{2}$ and $\\circled{3}$ are equivalences. As $\\ensuremath{\\catsingle{U}}\\subset\\ensuremath{\\catsingle{O}}(W)$ is a complete Weiss $\\infty$-cover, the map $\\circled{3}$ is an equivalence by \\cref{prop:descent}. To prove that $\\circled{2}$ is an equivalence, we show that for all $U\\in\\ensuremath{\\catsingle{U}}$ the individual maps before taking limits\n\\begin{equation}\\label{equ:emb-to-bimodule-map}\nE\\colon \\ensuremath{\\mathrm{Emb}}_\\partial(U,W')\\simeq \\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}}(U,W')\\longrightarrow \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q\\times I}}}(E_U,E_{W'})\n\\end{equation}\nare equivalences. To give a convincing proof of this, we rely on the specific construction of $E$ from \\cref{sec:the-functor} and refer to that section for the notation. Recall that the functor $E$ arose from restricting the codomain of the composition of simplicial objects in $\\infty$-categories\n\\begin{equation}\\label{equ:egeo-composition-embcalcproof}\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d) \\xlra{E^{\\mathrm{geo}}} \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)\\xrightarrow{(\\iota^*\\circ y)_*} \\overline{\\ensuremath{\\mathrm{ALG}}}(\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d))\\end{equation}\nwhere $(\\iota^*\\circ y)\\colon \\ensuremath{\\icat{M}\\mathrm{an}}_d\\rightarrow \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ is the Yoneda embedding followed by restriction along the inclusion $\\iota\\colon \\ensuremath{\\icat{D}\\mathrm{isc}}_d \\hookrightarrow \\ensuremath{\\icat{M}\\mathrm{an}}_d$. This factorisation induces a commutative diagram\n\\[\\begin{tikzcd}\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}\\dar{\\mathrm{inc}}\\rar{E^{\\mathrm{geo}}}&[10pt]\\ensuremath{\\mathrm{Bimod}}_{E^{\\mathrm{geo}}(P),E^{\\mathrm{geo}}(Q)}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)\\dar{U_{E^{\\mathrm{geo}}(P),E^{\\mathrm{geo}}(Q)}}\\rar{(\\iota^*\\circ y)_*}&[10pt] \\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}\\dar{U_{E_P,E_Q}}\\\\\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{[1]}\\rar &\\ensuremath{\\icat{M}\\mathrm{an}}_d\\rar{\\iota^*\\circ y}&\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d).\n\\end{tikzcd}\\]\nwhere the top composition is obtained from \\eqref{equ:egeo-composition-embcalcproof} by evaluation at $[1]$ and taking fibres of the face maps $(d_0,d_1)$, the middle and rightmost vertical map are the forgetful maps from \\cref{lemma:free-modules} and the bottom left horizontal map is the coherent nerve of the functor $\\ensuremath{\\cat{ncBord}}(d)_{[1]}\\rightarrow \\ensuremath{\\cat{Man}}_d$ of $\\ensuremath{\\cat{Kan}}$-enriched categories that sends a $[1]$-walled manifold $(W,\\mu)$ to $W|_{(\\mu(0)-\\epsilon,\\mu(1)+\\epsilon)}$. In particular, for $U=c(P)\\cup D \\cup c(Q)\\in\\ensuremath{\\catsingle{U}}$ considered as an object in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$, the inclusion $D\\subset U$ viewed as a morphism in $\\ensuremath{\\icat{M}\\mathrm{an}}_d$ gives a morphism $D\\rightarrow U_{E^{\\mathrm{geo}}(P),E^{\\mathrm{geo}}(Q)}(E^{\\mathrm{geo}}(U))$ in $\\ensuremath{\\icat{M}\\mathrm{an}}_d$, so by adjunction a morphism $F_{E^{\\mathrm{geo}}(P),E^{\\mathrm{geo}}(Q)}(D)\\rightarrow E^{\\mathrm{geo}}(U)$ in $\\ensuremath{\\mathrm{Bimod}}_{E^{\\mathrm{geo}}(P),E^{\\mathrm{geo}}(Q)}(\\ensuremath{\\icat{M}\\mathrm{an}}_d)$ which we claim to be an equivalence. By \\cref{lemma:free-modules} \\ref{enum:free-modules-iii}, it suffices to show that the image \n\\[U_{E^{\\mathrm{geo}}(P),E^{\\mathrm{geo}}(Q)}(F_{E^{\\mathrm{geo}}(P),E^{\\mathrm{geo}}(Q)}(D))\\longrightarrow U_{E^{\\mathrm{geo}}(P),E^{\\mathrm{geo}}(Q)}(E^{\\mathrm{geo}}(U))=U=c(P)\\cup D\\cup c(Q)\\]\nunder $U_{E^{\\mathrm{geo}}(M),E^{\\mathrm{geo}}(N)}$ is an equivalence. This is a consequence of the second part of \\cref{lemma:free-modules} \\ref{enum:free-modules-i}. Applying $(\\iota^*\\circ y)$ and using \\cref{lemma:free-modules} \\ref{enum:free-modules-iv}, it follows that the natural map $F_{E_{P \\times I},E_{Q \\times I}}(E_D)\\rightarrow E_U$ in $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}$ is an equivalence. As $F_{E_{P \\times I},E_{Q \\times I}}$ is left-adjoint to the forgetful functor $U_{E_{P \\times I},E_{Q \\times I}}$, the map \\eqref{equ:emb-to-bimodule-map} thus has the form\n\\[\n\t\\ensuremath{\\mathrm{Emb}}_\\partial(c(M)\\cup D\\cup c(N),W')=\\ensuremath{\\mathrm{Emb}}_\\partial(U,W')\\longrightarrow \\mathrm{Map}_{\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)}(E_D,E_{W'}).\n\\]\nand is given by the restriction map $\\ensuremath{\\mathrm{Emb}}_\\partial(c(M)\\cup D\\cup c(N),W')\\rightarrow \\ensuremath{\\mathrm{Emb}}_\\partial(D,W')$ followed by the map induced by the Yoneda embedding. The former is an equivalence by the contractibility of the space of collars and the latter is an equivalence by the Yoneda lemma since $D$ lies in the full subcategory $\\ensuremath{\\icat{D}\\mathrm{isc}}_d\\subset \\ensuremath{\\icat{M}\\mathrm{an}}_d$, so the composition is an equivalence.\n\\end{proof}\n\n\\begin{rem}\\label{rem:initial-among-rep-presheaves}\nThe first part of the previous proof in particular shows that for bordisms $W\\in\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}_{P,Q}$ that are diffeomorphic, relative to the ends, to $[0,1)\\times P\\sqcup T\\times\\ensuremath{\\mathbf{R}}^d \\sqcup (-1,0]\\times Q$ for some finite set $T$, the map \\eqref{equ:emb-calc} is an equivalence for all bordisms $W'\\in\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}_{P,Q}$. In particular, for $T=\\varnothing$, we see from the contractibility of the space of collars that both the source and target of this map are both contractible.  \n\\end{rem}\n\nCombining \\cref{thm:emb-calc} with the convergence of embedding calculus in handle codimension $\\ge3$ due to Goodwillie, Klein, and Weiss (see \\cite[Fact 5.1]{GoodwillieWeiss} and \\cite{GoodwillieKlein}), we conclude:\n\n\\begin{cor}\\label{cor:convergence} Fix bordisms $W,W'\\in\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$. If $W$ can be obtained from a closed collar of $P\\sqcup Q\\cong \\partial W$ by attaching handles of index $\\le d-3$, then the map\n\\[\n\t\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')\\simeq\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}}(W,W')\\rightarrow \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}}(E_W,E_{W'})\\simeq T_{\\infty}\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')\n\\]\ninduced by $E$ is an equivalence.\n\\end{cor}\n\n\\subsubsection{Comparison with the model of Boavida de Brito--Weiss'}\\label{sec:comparison-to-pedromichael}\\cref{thm:emb-calc} shows that the map \\eqref{equ:emb-calc} is a model for embedding calculus, so agrees up to weak equivalence with any other model Among the previously established models, that of Boavida de Brito--Weiss \\cite{BdBWSheaf} is closest to ours. Like ours, their model enhances the embedding calculus approximation $\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')\\rightarrow T_\\infty\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')$ to a functor on $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$. This section serves to extend \\cref{thm:emb-calc} to a comparison of the \\emph{functors} as opposed to just the individual maps on mapping spaces. This will in particular show that the monoid structures on $T_\\infty\\ensuremath{\\mathrm{Emb}}_\\partial(W,W)$ induced by composition in our and their model agree, which we will use in \\cref{sec:conf-cats}.\n\nFor this, we write $(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)_{P,Q}\\subset \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$ for the full subcategory of those bordisms that are diffeomorphic relative to the boundary to $P\\times[0,1)\\sqcup T\\times\\ensuremath{\\mathbf{R}}^d\\sqcup (-1,0]\\times Q$ for some finite set $T$. When translated from $\\ensuremath{\\cat{Kan}}$-enriched categories to $\\infty$-categories, Boavida de Brito's model for the embedding calculus approximation $\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')\\rightarrow T_\\infty\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')$ is the map on mapping spaces between $W$ and $W'$ induced by the composition\n\\[\n\t\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}\\xlra{y} \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q})\\xlra{\\iota^*}\\ensuremath{\\mathrm{PSh}}((\\ensuremath{\\icat{D}\\mathrm{isc}}_d)_{P,Q})\n\\] \nof the Yoneda embedding with the inclusion  $\\iota\\colon (\\ensuremath{\\icat{D}\\mathrm{isc}}_d)_{P,Q}\\hookrightarrow \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$ (cf.\\,Section 9 loc.cit.).\n\n\\begin{prop}\\label{prop:comparison-to-pedromichael}\nThere exists a comparison functor \n\\[\n\t\\mathrm{comp}\\colon\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}\\longrightarrow \\ensuremath{\\mathrm{PSh}}((\\ensuremath{\\icat{D}\\mathrm{isc}}_d)_{P,Q})\n\\] \nwhich fits into a commutative diagram of $\\infty$-categories\n\\[\\begin{tikzcd}[row sep=0.2cm]\n\t&\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}\\arrow[dl,\"E\",swap]\\arrow[dr,\"\\iota^*\\circ y\"]&\\\\\n\t\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}\\arrow[rr,\"\\mathrm{comp}\",swap]&& \\ensuremath{\\mathrm{PSh}}((\\ensuremath{\\icat{D}\\mathrm{isc}}_d)_{P,Q})\n\\end{tikzcd}\\]\nMoreover, the functor $\\mathrm{comp}$ becomes an equivalence after restricting its source and target to the essential images of the diagonal functors.\n\\end{prop}\n\n\\begin{proof}The functor $\\mathrm{comp}$ is defined as the  composition\n\\vspace{-0.1cm}\n\\[\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}} \\xlra{y} \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}})\\xlra{E^*} \\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q})\\xlra{\\iota^*}\n\t\\ensuremath{\\mathrm{PSh}}((\\ensuremath{\\icat{D}\\mathrm{isc}}_d)_{P,Q}) \\]\nWith this choice, the canonical natural transformation $y\\rightarrow E^*\\circ y \\circ E$ induces a natural transformation from the right-hand diagonal functor $(\\iota^*\\circ y)$ in the claimed triangle to $(\\mathrm{comp}\\circ E)$, so to prove the first claim it suffices to show that this is an equivalence. On a bordism $W\\in \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$ this natural transformation is the map of presheaves on $(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)_{P,Q}$ \n\\[\\ensuremath{\\mathrm{Emb}}_\\partial(-,W)\\simeq \\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}}(-,W)\\longrightarrow\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}}(E_{(-)},E_W)\\]\ninduced by $E$, which is an equivalence by \\cref{rem:initial-among-rep-presheaves}. To show the second claim, it suffices to show that for bordisms $W,W'\\in \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$, the bottom map in the commutative triangle in $\\ensuremath{\\catsingle{S}}$\n\\[\\begin{tikzcd}[row sep=0.3cm]\n\t&\\ensuremath{\\mathrm{Emb}}_\\partial(W,W')\\arrow[dl,\"E\",swap]\\arrow[dr,\"\\iota^*\\circ y\"]&\\\\\n\t\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}}(E_W,E_{W'})\\arrow[rr,\"\\mathrm{comp}_*\"] &&\\mathrm{Map}_{\\ensuremath{\\mathrm{PSh}}((\\ensuremath{\\icat{D}\\mathrm{isc}}_d)_{P,Q})}(E'_W,E'_{W'})\n\\end{tikzcd}\\]\nis an equivalence; here $E'_{(-)}$ is short for $(\\iota^*\\circ y)$. If $W$ is contained in $(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)_{P,Q}$, then both vertical maps are equivalences: the left one by \\cref{rem:initial-among-rep-presheaves} and the right one by the Yoneda lemma. By an argument similar to that in the proof of \\cref{thm:emb-calc}, it thus suffices to show that the two $\\ensuremath{\\catsingle{S}}$-valued functors $\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{M \\times I},E_{N \\times I}}}(E_{(-)},E_{W'})$ and $\\mathrm{Map}_{\\ensuremath{\\mathrm{PSh}}((\\ensuremath{\\icat{D}\\mathrm{isc}}_d)_{P,Q})}(E'_{(-)},E'_{W'})$ on $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$ satisfy descent for complete Weiss $\\infty$-covers. For the former this is \\cref{prop:descent} and for the latter it follows from a very similar argument; see also the proof of \\cite[Lemma 6.7]{KnudsenKupers}.\n\\end{proof}\n\n\\begin{rem}\nConsidering bordisms $W\\colon P\\leadsto Q$ as bordisms $\\varnothing \\leadsto P\\sqcup Q$ or $P\\sqcup Q\\leadsto \\varnothing$ leads to equivalences between $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}$, $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{\\varnothing,P\\sqcup Q},$ and $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P\\sqcup Q,\\varnothing}$, and similarly for $(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)_{P\\sqcup Q}$---compatible with $\\iota^*\\circ y$. It thus follows from \\cref{prop:comparison-to-pedromichael} that $E\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,Q}\\rightarrow  \\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{Q \\times I}}$ agrees up to equivalences and after restricting the target to the essential image with the analogous functors involving $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{\\varnothing},E_{P\\times I\\sqcup Q \\times I}}$ or $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P\\times I\\sqcup Q \\times I,E_{\\varnothing}}}$. That the latter two categories are equivalent can also be deduced from \\cref{rem:lurie-bimodules} and \\cite[4.6.3.11]{LurieHA} (no $(-)^\\mathrm{rev}$ appears since we secretly used the anti-homomorphism of $E_{P \\times I}$ or $E_{Q \\times I}$ by reflection in $I$).\n\\end{rem}\n\n\\subsection{Isotopy extension for $E$}\\label{sec:isotopy-extension}\nA key input in the proof of \\cref{bigthm:2-type-invariance} in \\cref{sec:2-type-invariance-sdisc} will be a version of the isotopy extension theorem for the mapping spaces in $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{P,Q}$.  In view of \\cref{thm:emb-calc}, this amounts to an isotopy extension theorem for embedding calculus. Such a theorem has been proved by Knudsen--Kupers \\cite[Theorem 6.1]{KnudsenKupers}, but instead of reducing the version we need from theirs, it is more convenient to give a direct proof based on their strategy.\n\nThe setting is as follows. We fix two compact bordisms $W\\colon P\\leadsto Q$,  $W'\\colon R\\leadsto S$, two possibly noncompact bordisms $M,N\\colon Q\\leadsto R$, and an open collar neighbourhood $c(M)\\subset M$ viewed as a bordism $Q\\leadsto R$. \nWriting $c$ for the inclusion $c(M)\\subset M$ viewed as a morphism in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{Q,R}$, we have a commutative diagram\n\\[\n\\begin{tikzcd}[column sep=1.5cm,ar symbol\/.style = {draw=none,\"\\textstyle#1\" description,sloped},\n\tequivalent\/.style = {ar symbol={\\simeq}}]\n\t\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{Q,R}}\\big(M,N\\big)\\dar{(-)\\circ c}\\rar{W\\cup_Q(-)\\cup_{R}W'}\\dar\\arrow[d, phantom, shift left=3cm, \"\\circled{$1$}\"]& \\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,S}}\\big(W\\cup_QM\\cup_RW',W\\cup_QN\\cup_RW'\\big)\\dar{(-)\\circ {(W\\cup_Qc\\cup_R{W'})}}\\\\\n\t\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{Q,R}}\\big(c(M),N\\big)\\arrow[d,equivalent]\\rar{W\\cup_Q(-)\\cup_{R}W'} &\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,S}}\\big(W\\cup_Qc(M)\\cup_RW',W\\cup_QN\\cup_RW'\\big)\\\\[-15pt]\n\t\\ast&&\n\\end{tikzcd}\\vspace{-0.2cm}\n\\]\nwhich maps via the functor $E\\colon \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)\\rightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)$ to the corresponding square for $\\ensuremath{\\icat{M}\\mathrm{od}}(d)$ \n\\[\\hspace{-0.3cm}\n\\begin{tikzcd}[column sep=2.5cm,ar symbol\/.style = {draw=none,\"\\textstyle#1\" description,sloped},\n\tequivalent\/.style = {ar symbol={\\simeq}}]\n\t\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q\\times I},E_{R\\times I}}}\\big(E_M,E_N\\big)\\dar{(-)\\circ E_c}\\arrow[d, phantom, shift left=3cm, \"\\circled{$2$}\"]\\rar{E_{W}\\cup_{E_{Q\\times I}}(-)\\cup_{E_{R\\times I}}E_{W'}}\\dar& \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P\\times I},E_{S\\times I}}}\\big(E_{W\\cup_QM\\cup_RW'},E_{W\\cup_QN\\cup_RW'}\\big)\\dar{(-)\\circ {E_{W\\cup_Qc\\cup_R{W'}}}}\\\\\n\t\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q\\times I},E_{R\\times I}}}\\big(E_{c(M)},E_N\\big)\\arrow[d,equivalent]\\rar{E_{W}\\cup_{E_{Q\\times I }}(-)\\cup_{E_{R\\times I}}E_{W'}} &\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P\\times I},E_{S\\times I}}}\\big(E_{W\\cup_Qc(M)\\cup_RW'},E_{W\\cup_QN\\cup_RW'}\\big)\\\\[-15pt]\n\t\\ast&\n\\end{tikzcd}\\vspace{-0.2cm}\n\\]\nNote that the bottom left corners in both squares are contractible by \\cref{rem:initial-among-rep-presheaves}. Moreover, in view of \\cref{lem:mapping-space-emb} the square $\\circled{1}$ has up to equivalence the form\n\\[\\begin{tikzcd}[ar symbol\/.style = {draw=none,\"\\textstyle#1\" description,sloped},\n\tequivalent\/.style = {ar symbol={\\simeq}}, column sep=1.5cm]\n\t\\ensuremath{\\mathrm{Emb}}_\\partial\\big(M,N\\big)\\dar{(-)\\circ c}\\rar{W\\cup_Q(-)\\cup_{R}W'}\\dar&[10pt]\\ensuremath{\\mathrm{Emb}}_\\partial\\big(W\\cup_QM\\cup_RW',W\\cup_QN\\cup_RW'\\big)\\dar{(-)\\circ {(W\\cup_Qc\\cup_R{W'})}}\\\\\n\t\\ensuremath{\\mathrm{Emb}}_\\partial\\big( c(M),N\\big)\\arrow[d,equivalent]\\rar{W\\cup_Q(-)\\cup_{R}W'} &\\ensuremath{\\mathrm{Emb}}_\\partial\\big(W\\cup_Qc(M)\\cup_RW',W\\cup_QN\\cup_RW'\\big)\\\\[-15pt]\n\t\\ast&\n\\end{tikzcd}\\vspace{-0.2cm}\\]\nAs the restriction map $\\ensuremath{\\mathrm{Emb}}_\\partial(W\\cup_Qc(M)\\cup_RW',W \\cup_Q N\\cup_R W')\\rightarrow \\ensuremath{\\mathrm{Emb}}_{P\\sqcup S}(W\\sqcup W',W\\cup_Q N\\cup_R W')$ is an equivalence and $W\\sqcup W'$ is compact, it follows from the parametrised isotopy extension theorem that this square is cartesian, so the same holds for the square $\\circled{1}$. \n\nThe isotopy extension result we will prove says that the same holds for $\\circled{2}$ under a certain condition on the convergence of embedding calculus, namely that the map from the bottom right of $\\circled{1}$ to the bottom right corner of $\\circled{2}$ is an equivalence if $M$ is replaced by $C_k\\coloneqq c(M)\\sqcup \\ul{k} \\times \\ensuremath{\\mathbf{R}}^d\\in \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{Q,R}$ for $\\ul{k} = \\{1,2,\\ldots,k\\}$ and all $k\\ge0$. We denote by $\\circled{2}^\\simeq$ the square obtained from $\\circled{2}$ by replacing the categories $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q \\times I},E_{R \\times I}}$ and $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{S\\times I}}$ in the top row by their cores. \n\n\\begin{thm}\\label{thm:isotopy-extension}\nIf the map induced by $E$\n\\[\\hspace{-0.1cm}\n\t\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P,S}}\\big(W\\cup_QC_k\\cup_RW',W\\cup_QN\\cup_RW'\\big)\\rightarrow\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I},E_{S\\times I}}}\\big(E_{W\\cup_QC_k\\cup_RW'},E_{W\\cup_QN\\cup_RW'}\\big)\n\\]\nis an equivalence for all $k\\ge0$, then the square $\\circled{2}$ is cartesian. If this assumption in addition holds for $M$ in place of $N$, then the square $\\circled{2}^\\simeq$ is also cartesian.\\end{thm}\n\n\\begin{proof}\nWe first show the claim for $\\circled{2}$. We write $\\ensuremath{\\catsingle{U}}$ for the poset of open subsets of $M$ that are unions $U=D\\cup c'(M)$ such that $c(M)\\subset M$ is an open collar of the boundary that contains the chosen collar $c(M)\\subset M$ and $D\\subset M$ is diffeomorphic to $T\\times \\ensuremath{\\mathbf{R}}^d$ for some finite set $T$. Considering $U$ as an object in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{Q,R}$ we have a functor $\\ensuremath{\\catsingle{U}}\\rightarrow \\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{Q,R}$. Since the square $\\circled{2}$ is natural in $M$, it maps to the limit of the same squares for $M$ replaced by $U\\in\\ensuremath{\\catsingle{U}}$\n\\[\\begin{tikzcd}\n\t\\underset{U\\in\\ensuremath{\\catsingle{U}}}{\\lim\\ }\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q \\times I,R\\times I}}}\\big(E_U,E_{N}\\big)\\dar\\rar\\dar&\\underset{U\\in\\ensuremath{\\catsingle{U}}}{\\lim\\ }\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I,S\\times I}}}\\big(E_{W\\cup_QU\\cup_{R}W'},E_{W\\cup_QN\\cup_{R}W'}\\big)\\dar\\\\\n\t\\underset{U\\in\\ensuremath{\\catsingle{U}}}{\\lim\\ }\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q \\times I,R\\times I}}}\\big(E_{c(M)},E_{N}\\big)\\rar &\\underset{U\\in\\ensuremath{\\catsingle{U}}}{\\lim\\ }\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times I,S\\times I}}}\\big(E_{W\\cup_Qc(M)\\cup_RW'},E_{W\\cup_QN\\cup_RW'}\\big).\n\\end{tikzcd}\\]\nWe claim that it suffices to show this square of limits is cartesian. To justify this, we show that the maps from $\\circled{2}$ to the square of limits are all equivalences. For the maps between the bottom left corners and between the bottom right corners, this follows from the fact that the diagram is constant and the category $\\ensuremath{\\catsingle{U}}$ is weakly contractible since it is cofiltered. For the top-right corner and top-left corner it follows from \\cref{prop:descent} since the posets $\\ensuremath{\\catsingle{U}}$ and $\\{W\\cup_QU\\cup_RW'\\,|\\,U\\in\\ensuremath{\\catsingle{U}}\\}$ are complete Weiss $\\infty$-covers of $M$ and $W\\cup_QM\\cup_RW'$. \n\t\nTo show that the previous square of limits is cartesian, note that it receives a map from the analogous square using $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ instead of $\\ensuremath{\\icat{M}\\mathrm{od}}(d)$, and this map of squares consists of equivalences: for the top right and bottom right corner it holds by assumption and for the top left and bottom left corner it holds by \\cref{rem:initial-among-rep-presheaves}. The square using $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$ is a limit of squares of the form $\\circled{1}$, with $M$ replaced by $U\\in\\ensuremath{\\catsingle{U}}$, so it is cartesian since we have already explained that $\\circled{1}$ is cartesian and limits of cartesian squares remain cartesian.\n\t\nTo show the claim for ${\\circled{2}}^\\simeq$, we first assume $M=N$ in which case, the claim follows from the following fact: given a monoid $A$ in $\\ensuremath{\\catsingle{S}}$ acting on a space $X$ and $x\\in X$, consider the fibre sequence\n\\[\n\t\\mathrm{hofib}_{x}(A\\xlra{(-)\\cdot x}X)\\longrightarrow A\\xlra{(-)\\cdot x}X \n\\]\nwhose fibre inherits the structure of a monoid in $\\ensuremath{\\catsingle{S}}$ from that of $A$. Then the sequence obtained by passing to group-like components in fibre and total space is again a fibre sequence. This follows from the long exact sequence of the original fibre sequence, using that in an exact sequence of monoids $A_0\\rightarrow A_1\\rightarrow A_2$, the monoid $A_1$ is a group if $A_0$ and $A_2$ are.\n\nTo deduce the general case of ${\\circled{2}}^\\simeq$ from that of ${\\circled{2}}$, it suffices to prove that if $\\varphi \\colon E_M \\to E_N$ has the property that $\\varphi' \\coloneqq \\mathrm{id}_{E_W} \\cup_{E_{Q \\times I}}  \\varphi \\cup_{E_{R \\times I}} \\mathrm{id}_{E_{W'}}$ is an equivalence, then $\\varphi$ is also an equivalence. To prove this, we pick an inverse $\\psi' \\colon E_{W \\cup_Q N \\cup_R W'} \\to E_{W \\cup_Q M \\cup_R W'}$ to $\\varphi'$ and claim that the image of $\\psi'$ under the right-vertical map in the square ${\\circled{2}}$ with the role of $M$ and $N$ reversed lies in the component of the bottom horizontal map. To see this, we extend this square to the bottom by\n\\[\\begin{tikzcd} \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q\\times I},E_{R\\times I}}}\\big(E_{c(M)},E_N\\big) \\rar \\dar{\\varphi\\circ(-)}[swap]{\\simeq} & \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P\\times I},E_{S\\times I}}}\\big(E_{W\\cup_Q c(M)\\cup_RW'},E_{W\\cup_QN\\cup_RW'}\\big) \\dar{\\varphi' \\circ -}[swap]{\\simeq} \\\\\n\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q\\times I},E_{R\\times I}}}\\big(E_{c(M)},E_M\\big) \\rar & \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P\\times I},E_{S\\times I}}}\\big(E_{W\\cup_Q c(M) \\cup_RW'},E_{W\\cup_Q M\\cup_RW'}\\big)\\end{tikzcd}\\]\nwhere the left vertical map is an equivalence as both source and target are contractible, and the right vertical map is an equivalence because $\\varphi'$ is one by assumption. To see whether the image of $\\psi'$ in the upper right corner is in the component hit by the upper horizontal map, it thus suffices to show that the image of $\\psi'$ in the bottom horizontal corner is in the component hit by the bottom horizontal map. But this follows from the relation $[\\varphi'\\circ\\psi']=[\\mathrm{id}]$ in the set of components, which holds by the choice of $\\psi'$. Using that the square ${\\circled{2}}$ with the role of $M$ and $N$ reverse is a pullback (this is where we use the additional hypothesis for $M$), we conclude that there exists $\\psi \\colon E_N \\to E_M$ such that $[\\psi']= [W\\cup_Q\\psi\\cup_RW']$. To finish the proof, it suffices to show that $\\varphi\\circ\\psi$ and $\\psi\\circ\\varphi$ are both equivalences, since then $\\varphi$ has to be an equivalence. But this follows from the case $M=N$ treated above, using that both compositions become equivalences after applying $W\\cup_Q(-)\\cup_RW'$ since this even holds for $\\psi$ and $\\varphi$ individually.\n\\end{proof}\n\n\\begin{rem}The proof of \\cref{thm:isotopy-extension} in particular shows that if the assumption in the statement holds for $M$ and $N$, then the following map detects equivalences:\n\\vspace{-0.05cm}\n\\[\n\t\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q\\times I},E_{R\\times I}}}\\big(E_M,E_N\\big) \\xrightarrow{E_{W}\\cup_{E_{Q\\times I}}(-)\\cup_{E_{R\\times I}}E_{W'}} \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P\\times I},E_{S\\times I}}}\\big(E_{W\\cup_QM\\cup_RW'},E_{W\\cup_QN\\cup_RW'}\\big)\n\\]\n\\end{rem}\n\n\\subsection{$\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces}\\label{sec:disc-structure-spaces} We conclude this section with the definition of the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces and a discussion some of their functoriality. Given objects $P\\in \\ensuremath{\\icat{B}\\mathrm{ord}}(d)$ and $A\\in\\ensuremath{\\icat{M}\\mathrm{od}}(d)$, i.e.\\,a closed $(d-1)$-manifold $P$ and an associative algebra $A$ in $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$, we abbreviate the $\\infty$-category of nullbordisms of $P$ and the analogue for right $A$-modules by\n\\begin{equation}\\label{equ:abbreviate-right-modules}\n\t\\gls*{nullbordism} \\coloneqq \\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{\\varnothing,P} \\qquad \\text{and} \\qquad \\gls*{rightbordism} \\coloneqq \\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_\\varnothing,A}.\n\\end{equation}\n\n\\begin{rem}Note that $E_\\varnothing$ is the monoidal unit in $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$, so $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_\\varnothing,A}$ may be viewed as an $\\infty$-category of right-$A$-modules. Using \\cref{rem:lurie-bimodules} and \\cite[4.3.2.8]{LurieHA} one sees that this agrees with Lurie's model of the $\\infty$-category of right-$A$-modules, but we will not use this.\\end{rem}\n\n\\subsubsection{$\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces of modules}For $A=E_{P\\times I}$ for a closed $(d-1)$-manifold $P$, the functor $E$ induces a functor $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P}\\rightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)_{P\\times I}$. As the source is an $\\infty$-groupoid by the discussion \\cref{sec:details-bord}, it lands in the core $\\ensuremath{\\icat{M}\\mathrm{od}}(d)^\\simeq_{P\\times I}\\subset \\ensuremath{\\icat{M}\\mathrm{od}}(d)_{P\\times I}$. The $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces are the fibres of this functor of $\\infty$-groupoids:\n\n\\begin{dfn}\\label{def:disc-structure-space} The \\emph{$\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space} of a right-$E_{P \\times I}$-module $X\\in\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P\\times I}}$ is the fibre \\[\\gls*{sdisc}\\coloneqq \\mathrm{fib}_{X}(\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P}\\rightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{E_{P \\times I}})\\in \\ensuremath{\\catsingle{S}}\\]\n\\end{dfn}\nFrom the description of the object and mapping spaces of $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)$ and $\\ensuremath{\\icat{M}\\mathrm{od}}(d)$ in \\cref{sec:mapping-infinity-category}, we see that the path components of $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_P(X)$ are given by\n\\[\n\t\\pi_0\\,S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_P(X) = \\frac{\\left\\{\\text{\\parbox{11.5cm}{\\centering pairs $(M,\\varphi)$ of a compact smooth $d$-manifold $M$ with identified boundary $\\partial M\\cong P$ \\newline and an equivalence of right $E_{P \\times I}$-modules $\\varphi \\colon E_M \\to X$}}\\right\\}}{\\parbox{9.5cm}{\\centering $(M,\\varphi) \\sim (M',\\varphi')\\Leftrightarrow$ there exists a diffeomorphism $\\alpha \\colon M \\to M'$ relative to $P$ with $[\\varphi' \\circ E_\\alpha]=[\\varphi]\\in\\pi_0\\,\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{E_{P \\times I}}}(E_M,X)$}}\n\\] \nand that the component of a pair $(M,\\varphi)$ agrees with the identity component\n\\[\n\tS^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_P(X)_{(M,\\varphi)} \\simeq \\big(\\mathrm{Aut}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{P\\times I}}(E_M)\/\\ensuremath{\\mathrm{Diff}}_\\partial(M)\\big)_{\\mathrm{id}}.\n\\]\nof the fibre $\\mathrm{Aut}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_P}(E_M)\/\\ensuremath{\\mathrm{Diff}}_\\partial(M)$ of the map $\\ensuremath{\\mathrm{BDiff}}_\\partial(M)\\rightarrow \\mathrm{BAut}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_P}(E_M)$ induced by $E$. This can also be rephrased in the form of an equivalence\n\\begin{equation}\\label{equ:disjoint-union-description-sdisc}\n\t \\textstyle{S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_P(X)\\simeq \\bigsqcup_{[M]}\\mathrm{Aut}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{P\\times I}}(E_M)\/\\ensuremath{\\mathrm{Diff}}_\\partial(M)}\n\\end{equation}\nwhere $[M]$ runs through diffeomorphism classes of compact manifolds $M$ with identified boundary $\\partial M\\cong P$ for which there exists an equivalence $E_M\\rightarrow X$ of right $E_{P\\times I}$-modules.\n\n\n\\subsubsection{$\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces of manifolds}\\label{sec:sdisc-for-manifolds}\nGiven a compact $d$-manifold $W$ with identified boundary $\\partial W\\cong P$, considered as an object in $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P}$, we abbreviate \\[\\gls*{sdiscpartial} \\coloneqq S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_{P}(E_W).\\] \nThis is natural in $\\smash{W \\in \\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}_{\\varnothing\/}}$ in that it gives a functor $\\smash{S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(-) \\colon \\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}_{\\varnothing\/} \\rightarrow \\ensuremath{\\catsingle{S}}}$ from the $\\infty$-category of nullbordisms to the $\\infty$-category of spaces. In particular, for bordisms $W\\colon\\varnothing\\leadsto P$ and $W' \\in\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P,Q}$ we have a \\emph{gluing map} $(- \\cup_P W) \\colon S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(W) \\rightarrow S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(W \\cup_P W')$.\n\n\n\\section{\\cref{bigthm:2-type-invariance}: $2$-type invariance} \\label{sec:2-type-invariance}\nThe goal of this section is to prove \\cref{bigthm:2-type-invariance}, which says that the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space of a compact $d$-manifold depends for $d\\ge5$ only on the tangential 2-type, a notion that we recall in \\cref{sec:tangential-k-types}. As outlined in \\cref{sec:intr-2-type-invariance}, this will be an application of a general tangential $k$-type invariance result, proved in \\cref{sec:k-type-invariance}, about the values of certain functors on a  category of compact null bordisms. That $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(-)$ satisfies its hypotheses is verified in \\cref{sec:2-type-invariance-sdisc}.\n\n\\begin{nconvention}\\\n\\begin{enumerate}\n\\item In contrast to the previous sections, all manifolds---which were already assumed to be smooth---are now also assumed to be compact. Non-empty boundaries are allowed. \n\\item In this section we adopt the point of view on $\\theta$-structures in terms of bundle maps (always required to be fibrewise injective), which is different but by basic bundle theory equivalent to that in terms of $\\mathrm{GL}_d(\\ensuremath{\\mathbf{R}})$-spaces from \\cref{sec:details-tangential-bord}. For the convenience of the reader, we recall the necessary definitions from scratch in \\cref{sec:theta-manifolds}.\n\\end{enumerate}\\end{nconvention}\n\n\n\\subsection{Tangential $k$-types} \\label{sec:tangential-k-types} We start with some manifold-theoretic preliminaries.\n\n\\subsubsection{$\\theta$-manifolds and tangential $k$-types} \n\\label{sec:theta-manifolds}\nGiven a map $\\theta\\colon B\\rightarrow \\mathrm{BO}$, a \\emph{$\\theta$-manifold} $M$ is a manifold with a \\emph{$\\theta$-structure} on its stable tangent bundle, by which we mean in this section a stable bundle map $\\ell_M\\colon \\tau_M^s\\rightarrow \\theta^*\\gamma$ from the stable tangent bundle of $M$ to the pullback of the universal stable vector bundle $\\gamma$ over $\\mathrm{BO}$ along $\\theta$. A tangential structure is \\emph{$k$-connected} if the underlying map $\\bar{\\ell}_M\\colon M\\rightarrow B$ is $k$-connected in the usual sense. \n\nGiven a codimension $0$ embedding $e\\colon M\\hookrightarrow N$ and a $\\theta$-structure $\\ell_N$ on $N$, we obtain a $\\theta$-structure $e^*\\ell_N$ on $M$ by precomposition with the stable derivative of $e$. Two $\\theta$-manifolds $M$ and $N$ are \\emph{$\\theta$-diffeomorphic} if there exists a diffeomorphism $\\phi\\colon M\\rightarrow N$ of the underlying manifolds such that $\\phi^*\\ell_N$ and $\\ell_M$ are homotopic as bundle maps. A codimension $0$ embedding $e\\colon M\\hookrightarrow N$ is an \\emph{equivalence on tangential $k$-types} if $N$ admits a $k$-connected $\\theta$-structure $\\ell_N$ for some $\\theta$ such that $e^*\\ell_N$ is again $k$-connected. Two manifolds $M$ and $N$ have the \\emph{same tangential $k$-type} if there is a $\\theta\\colon B\\rightarrow \\mathrm{BO}$ such that  $M$ and $N$ admit $k$-connected $\\theta$-structures $\\ell_M$ and $\\ell_N$ (for the same $\\theta$).\n\n\\begin{ex}\\label{rem:2-connected-maps}\nAny codimension $0$ embedding $M\\hookrightarrow N$ that is $k$-connected is an equivalence on tangential $k$-types. This is clear from the definition as long as $N$ admits  a $k$-connected $\\theta$-structure with respect to \\emph{some} $\\theta$, and there is indeed always such a choice: pick a Moore-Postnikov factorisation $N\\rightarrow B\\rightarrow\\mathrm{BO}$ of a classifying map for the stable tangent bundle of $N$ into a $k$-connected map followed by a $k$-coconnected map $\\theta\\colon B\\rightarrow\\mathrm{BO}$.\n\\end{ex}\n\n\\begin{ex}\\label{rem:classification-2-types}The case of most interest to us is $k=2$, where there is a simple recipe to decide whether two $d$-manifolds $M_0$ and $M_1$ have the same tangential $k$-types. If the $M_i$ are disconnected, then they have the same tangential $2$-type if and only if there exists a bijection between their components such that the corresponding components have the same tangential $2$-type. For connected manifolds $M_0$ and $M_1$, one can decide whether they have the same tangential $2$-type as follows (cf.\\,\\cite[p.\\,712--713]{Kreck}; Kreck deals with \\emph{normal} $k$-types as opposed to \\emph{tangential} $k$-types and has a different indexing convention, but neither of this makes a difference):\n\t\\begin{enumerate}\n\t\t\\item The functionals $w_2(M_i) \\colon \\pi_2(M_i)\\rightarrow \\ensuremath{\\mathbf{Z}}\/2$ for $i=0,1$ induced by the second Stiefel--Whitney classes need to be both trivial or nontrivial.\n\t\t\\item If they are both nontrivial, then $M_0$ and $M_1$ have the same tangential $2$-type if and only if there exists an abstract isomorphism $\\varphi\\colon \\pi_1(M_0)\\rightarrow\\pi_1(M_1)$ such that $\\varphi ^*w_1(M_1)=w_1(M_0)$, where $w_1(M_i) \\in \\ensuremath{\\mathrm{H}}^1(M_i;\\ensuremath{\\mathbf{Z}}\/2)\\cong \\ensuremath{\\mathrm{H}}^1(K(\\pi_1M_i,1);\\ensuremath{\\mathbf{Z}}\/2)$ is the first Stiefel--Whitney class.\n\t\t\\item If they are both trivial, then there are unique classes $w_2(M_i) \\in \\ensuremath{\\mathrm{H}}^2(K(\\pi_1(M_i),1);\\ensuremath{\\mathbf{Z}}\/2)$ that pull back to the second Stiefel--Whitney classes along the canonical maps $M_i\\rightarrow K(\\pi_1(M_i),1)$. Then $M_0$ and $M_1$ have the same tangential $2$-type if and only if there exists an abstract isomorphism $\\varphi\\colon \\pi_1(M_0)\\rightarrow\\pi_1(M_1)$ with $\\varphi ^*w_i(M_1)=w_i(M_0)$ for $i=1,2$.\n\t\\end{enumerate}\nIn particular, if $M_0$ and $M_1$ are spin, $w_i(M)$ and $w_i(N)$ vanish for $i\\le2$, so the recipe shows that they have the same tangential $2$-types if and only if their fundamental groupoids are equivalent.\n\\end{ex}\n\n\\begin{lem}\\label{lem:nice-representative-k-type} \nLet $M$ be an $m$-manifold and $k\\ge 0$ a number. For any $d\\ge4$ with $k\\le \\lfloor \\tfrac{d}{2}\\rfloor$, there exists a closed $d$-manifold $P$ with the same tangential $k$-type as $M$.\n\\end{lem}\n\n\\begin{proof}\nWe may assume $k\\ge1$ and that $M$ is connected; apply the claim to each connected component otherwise. Choose a Moore--Postnikov $k$-factorisation $M\\rightarrow B\\rightarrow \\mathrm{BO}$ of the stable tangent bundle into a $k$-connected map followed by a $k$-coconnected map $\\theta\\colon B\\rightarrow \\mathrm{BO}$. The condition $k\\le \\lfloor \\tfrac{d}{2}\\rfloor$ in particular implies that $d \\ge k+1$, so the $d$-sphere $S^d$ admits a $\\theta$-structure by obstruction theory. Doing surgeries compatible with the $\\theta$-structure (see \\cite[Proposition\\,4]{Kreck}), we obtain a closed $d$-manifold $P$ with a $k$-connected $\\theta$-structure.\n\\end{proof}\n\n\\subsubsection{$\\theta$-bordism}\\label{sec:theta-manifolds}\nGiven a $\\theta$-manifold $M$, a choice of inwards pointing vector field induces a $\\theta$-structure on the boundary $\\partial M$. Using the canonical vector field $\\smash{\\frac{\\partial}{\\partial x}}$ on $[0,1]$, we moreover obtain a $\\theta$-structure on $M\\times [0,1]$, which restricts to a $\\theta$-structure on the \\emph{double} $M\\cup_{\\partial M}\\overline{M} \\cong \\partial(M\\times [0,1])$ of $M$. Here $\\overline{M}$ is the $\\theta$-manifold whose underlying manifold is $M$ but which is equipped with the \\emph{opposite $\\theta$-structure} obtained by restricting the induced $\\theta$-structure on $M\\times [0,1]$ to $M\\times\\{1\\}\\subset \\partial(M\\times [0,1])$. A \\emph{$\\theta$-bordism} from a $d$-dimensional $\\theta$-manifold $P$ to another $d$-dimensional $\\theta$-manifold $Q$ is a $(d+1)$-dimensional $\\theta$-manifold $W$ together with a $\\theta$-diffeomorphism $\\partial W\\cong P\\sqcup \\smash{\\overline{Q}}$; we denote this $W \\colon P \\gls*{bordism} Q$. A $\\theta$-manifold $P$ is \\emph{$\\theta$-null bordant} if there is a $\\theta$-bordism $P \\leadsto \\varnothing$. Note that, by construction, the double $M\\cup_{\\partial M} \\overline{M}$ of any $\\theta$-manifold $M$ is $\\theta$-nullbordant.\n\n\\subsubsection{Handle decompositions}\\label{sec:handle-dec}\nGiven a compact $d$-dimensional bordism $W\\colon P\\leadsto Q$ between closed $(d-1)$-manifolds, a \\emph{handle decomposition of the bordism} $W$ is a decomposition\n\\[\n\tP=W_{-1}\\overset{W(-1,0]}{\\leadsto}W_{0}\\overset{W(0,1]}{\\leadsto}\\cdots \\overset{W(d-2,d-1]}{\\leadsto}W_{d-1}\\overset{W(d-1,d]}{\\leadsto}W_{d}=Q\n\\]\nof $W$ as a union of bordisms between closed $(d-1)$-manifolds $W_i$ such that $W(k-1,k]$ is obtained from a collar on $W_{k-1}$ by attaching finitely many handles of index $k$. Such a decomposition always exists, for instance by choosing a self-indexing Morse function. By construction, $W_{k+1}$ is obtained from $W_{k}$ by finitely many $k$-surgeries. We abbreviate \n\\[\n\t\\gls*{whalf} \\coloneqq \\cup_{m\\le i \\le k-1}W(i,i+1] \\quad \\text{and} \\quad \\gls*{wfull} \\coloneqq \\cup_{m-1\\le i \\le k-1}W(i,i+1]\n\\]\nand consider these manifolds as bordisms from $W_m$ to $W_k$ and from $W_{m-1}$ to $W_k$, respectively. The idea behind the notation is that the half-open or closed interval indicates which handles the submanifold contains. Given $m\\le k$, we say that $W$ has \\emph{handle type} $[m,k]$ if there is a handle decomposition with $W=W[m,k]$. A $d$-manifold $M$ \\emph{handle type $[m,k]$} if it has that property when viewed as a bordism $M\\colon\\varnothing \\leadsto \\partial M$. It is said to have \\emph{handle dimension $\\le k$} if it has handle type $[0,k]$. A codimension $0$ submanifold inclusion $N \\subset \\mathrm{int}(M)$ has \\emph{relative handle type $[m,k]$} if the bordism $M\\backslash \\mathrm{int}(N)\\colon \\partial N\\leadsto \\partial M$ has handle type $[m,k]$, and $N\\subset \\mathrm{int}(M)$ has \\emph{relative handle dimension $\\le k$} if this bordism has handle type $[0,k]$.\n\n\\subsubsection{Handle trading and connectivity}\n\nThe following two lemmas are certainly standard, but we could not find references for them in the generality we needed.\n\\begin{lem}\\label{lem:handle-trading} Let $W\\colon P\\leadsto Q$ be a bordism between closed $d$-manifolds $P$ and $Q$ with $d\\ge4$. If both boundary inclusions $P\\subset W\\supset Q$ are $k$-connected for some $k\\ge0$, then the following holds.\n\\begin{enumerate}\n\t\\item If $2k<d-1$, then $W\\colon P\\leadsto Q$ has handle type $[k+1,d-k]$ and \n\t\\item If $2k=d-1$, then $W\\sharp (S^{k+1}\\times S^{k+1})^{\\sharp r}\\colon P\\leadsto Q$ has type $[k+1,d-k]$ for some $r\\ge0$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}We begin with the first case. Starting from a handle decomposition of the bordism $W\\colon P\\leadsto Q$, we obtain a potentially different handle decomposition of $W$ of type $[k+1,d+1]$ by handle trading, without changing the number of $i$-handles for $i\\ge k+3$ (see e.g.\\,the proof of \\cite[Theorem 3]{WallConnectivity}). Now we apply the same procedure to the dual of this new handle decomposition to obtain yet another handle decomposition, this time of type $[0,d-k]$ and with the same number of $i$-handles for $i\\le d-k-2$. Since $k\\le d-k-2$ and we previously arranged that there are no $i$-handles for $i\\le k$, the resulting decomposition has type $[k+1,d-k]$.\n\t\nIn the case $2k=d-1$, we may reindex so that the claim reads as follows (set $n=k+1$): given a $2n$-dimensional bordism $W\\colon P\\leadsto Q$ with $2n\\ge6$ such that the inclusions $P\\subset W\\supset Q$ are $(n-1)$-connected, there exists an $r\\ge0$ such that the bordism $(W\\sharp (S^n\\times S^n)^{\\sharp r})\\colon P\\leadsto Q$ admits a handle decomposition with only $n$-handles. We are not aware of a classical reference for this fact; we learnt it from the proof of \\cite[Lemma 6.21]{GRWstable}.\n\\end{proof}\n\n\\begin{lem}\\label{lem:bordism-connectivity}\nLet $\\theta\\colon B\\rightarrow \\mathrm{BO}$ be a map and $(P,\\ell_P)$ and $(Q,\\ell_Q)$ two closed $d$-dimensional $\\theta$-manifolds that are $\\theta$-bordant. Assume $d\\ge4$ and fix $k\\ge0$ with $2k<d$.\n\\begin{enumerate}\n\t\\item If $\\ell_P$ is $k$-connected, then is a $\\theta$-bordism $W\\colon P\\leadsto Q$ such that $P\\subset W$ is $k$-connected.\n\t\\item If also $\\ell_Q$ is $k$-connected, then we may assume that $W\\supset Q$ is $k$-connected as well.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}For part (i), we refer to the proof of the correction \\cite[Proposition p.\\,48]{HebestreitJoachim} to a part of \\cite[Proposition 4]{Kreck}. The proof of part (ii) is a minor extension of their argument which we spell out for the convenience of the reader in the case $k\\ge1$, leaving $k=0$ as an easy exercise.\n\t\nStarting from a $\\theta$-bordism $(W,\\ell_W)\\colon (P,\\ell_P)\\leadsto (Q,\\ell_Q)$ we can assume that $\\ell_W$ is $k$-connected by performing surgery in the interior of $W$. As $\\ell_P$ and $\\ell_Q$ are $k$-connected, this ensures that the inclusions $P\\subset W\\supset Q$ induce an isomorphism on homotopy groups at all basepoints in degrees $\\le k-1$. By considering each component in $\\pi_0(P)\\cong\\pi_0(W)\\cong \\pi_0(Q)\\cong\\pi_0(B)$ separately we may assume that each of $P,W,Q,B$ is connected. We now consider the long exact sequences \n\\[\\begin{tikzcd}[column sep=0.5cm, row sep=0.1cm]\n\t\\ldots \\rar&\\begin{cases}\\pi_k(P)\\\\\\pi_k(Q)\\end{cases}\\rar[shorten <=-9pt]\\arrow[dr,two heads,shorten <=-4pt]& \\pi_k(W)\\arrow[d,two heads]\\arrow[r,two heads]&\\begin{cases}\\pi_k(W,P)\\\\\\pi_k(W,Q)\\end{cases}\\rar[shorten <=-9pt]{0}&\\begin{cases}\\pi_{k-1}(P)\\\\\\pi_{k-1}(Q)\\end{cases}\\rar[shorten <=-9pt]{\\cong}\\arrow[dr,\"\\cong\",swap,shorten <=-4pt]&\\pi_{k-1}(W)\\dar{\\cong}\\rar& \\ldots \\\\\n\t&&\\pi_k(B)&&&\\pi_{k-1}(B)&\n\\end{tikzcd}\\]\nof the pairs $(W,P)$ and $(W,Q)$. We first assume $k\\ge2$. By the relative Hurewicz theorem, we have $\\pi_k(W,P)\\cong \\ensuremath{\\mathrm{H}}_k(W,P)$ and similarly for $\\pi_k(W,Q)$, so these groups are in particular finitely generated. Contemplating the diagram shows that there are finite sets of elements $\\{p_i\\}$ and $\\{q_i\\}$ of $\\pi_k(W)$ that (i) map trivially to $\\pi_k(B)$ (and thus trivially to $\\pi_k(\\mathrm{BO})$) and (ii) map to sets of generators of $\\pi_k(W,P)$ and $\\pi_k(W,Q)$ respectively. As $2k<d$ we may represent these elements by two disjoint embeddings $\\overline{p}\\colon {\\sqcup^i} S^k\\times D^{d+1-k}\\hookrightarrow \\mathrm{int}(W)$ and $\\overline{q}\\colon {\\sqcup^i} S^k\\times D^{d+1-k}\\hookrightarrow \\mathrm{int}(W)$. Doing $\\theta$-surgery on these embeddings (see \\cite[Lemma 2]{Kreck}) yields a $\\theta$-bordism $W'\\colon P\\leadsto Q$ which we claim to satisfy the requirements of the statement, that is $\\pi_i(W',P)=0$ and  $\\pi_i(W',Q)=0$ for $i\\le k$. The reason being that (i) $\\pi_i(W',P)$ vanishes for $i\\le k-1$ since it is isomorphic to  $\\pi_i(W,P)=0$ and (ii) $\\pi_k(W',Q)$ vanishes since it is a quotient of $\\pi_k(W,Q)$ by a subgroup that contains the image of $\\{p_i\\}$ and $\\{q_i\\}$ and we chose the $\\{p_i\\}$ so that their image generates $\\pi_k(W',P)$. The same argument applies to the groups $\\pi_i(W',Q)$, so the claim in the case $k\\ge2$ follows. For $k=1$, the same argument applies even though $\\pi_1(W,P)$ and $\\pi_1(W,Q)$ need no longer be groups: instead of the relative Hurewicz theorem, one uses that $\\pi_1(W)$ is finitely generated, being the fundamental group of a compact manifold.\n\\end{proof}\n\nCombining the previous two lemmas we get:\n\n\\begin{cor}\\label{cor:theta-bordism-surgery}\n\tLet $\\theta\\colon B\\rightarrow \\mathrm{BO}$ be a map and $(P,\\ell_P)$ and $(Q,\\ell_Q)$ two closed $\\theta$-manifolds of dimension $d\\ge4$ that are $\\theta$-bordant. If $\\ell_P$ and $\\ell_Q$ are $k$-connected for some $k\\ge0$ with $2k<d$, then $Q$ can be obtained from $Q$ by a finite sequence of $p$-surgeries with $k\\le p\\le d-k-1$.\n\\end{cor}\n\n\\begin{proof}Lemmas~\\ref{lem:handle-trading} and \\ref{lem:bordism-connectivity} ensure that there is a bordism $W\\colon P\\leadsto Q$ of handle type $[k+1,d-k]$, which implies the statement.\n\\end{proof}\n\n\\subsection{$k$-type invariance}\\label{sec:k-type-invariance} \nTo state the announced tangential $k$-type result, we denote by $\\gls*{hmanc}$ the $1$-category whose objects are smooth compact $d$-manifolds (potentially with boundary) and whose morphisms are isotopy classes of codimension $0$ embeddings. Fixing another $1$-category $\\ensuremath{\\cat{C}}$, we prove the following result for functors of the form $F\\colon h\\ensuremath{\\cat{Man}^{\\mathrm{c}}}_d\\rightarrow \\ensuremath{\\cat{C}}$.\n\n\n\\begin{thm}\\label{thm:abstract-k-type-invariance}\nLet $d\\ge4$ and $F\\colon h\\ensuremath{\\cat{Man}^{\\mathrm{c}}}_d\\rightarrow \\ensuremath{\\cat{C}}$ a functor such that $F$ maps codimension $0$ submanifold inclusions of relative handle type $[k+1,d]$ to isomorphisms for some fixed $0\\le k<d\/2$. Then for any compact $d$-manifolds $M$ and $N$ of the same tangential $k$-type, the following holds \n\\begin{enumerate}\n\t\\item\\label{k-type-i} There exists an isomorphism $F(M)\\cong F(N)$.\n\t\\item\\label{k-type-ii} For any codimension $0$ embedding $e\\colon L\\hookrightarrow M$ where $L$ has handle dimension $\\le k$, there is an embedding $e'\\colon L\\hookrightarrow N$ for which the isomorphism \\ref{k-type-i} can be chosen so that the diagram\n\t\\[\\begin{tikzcd}[row sep=0.2cm]\n\t\t& F(L) \\arrow{ld}[swap]{F(e)} \\arrow{rd}{F(e')} & \\\\\n\t\tF(M) \\arrow{rr}{\\cong} & & F(N)\n\t\\end{tikzcd}\\]\n\tis commutative.\n\t\\item\\label{k-type-iii} If $k$ also satisfies $2\\le k<\\tfrac{d-1}{2}$, then any embedding $e\\colon M\\hookrightarrow N$ that is an equivalence on tangential $k$-types induces an isomorphism $F(M)\\cong F(N)$ as in \\ref{k-type-i}.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{rem}\\label{rem:psc-relation}\n\\cref{thm:abstract-k-type-invariance} is based on arguments we learnt from the literature on the space of positive scalar curvature metrics on a manifold $M$, in particular \\cite{EbertRWbordism,EbertWiemeler}. This space shares strong formal properties with the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space: it is often an infinite loop space \\cite[Theorems A-B]{EbertRWbordism}, depends conjecturally only on the tangential $2$-type (see \\cite[Conjecture C]{EbertWiemeler} and \\cite[Section 9]{EbertRWbordism}), and is often non-trivial (see e.g.~\\cite[Remark 1.1.1]{EbertRWbordism}).\n\\end{rem}\n\n\n\\begin{rem}\\label{rem:nulbbordism-cat}\nTaking complements, $h\\ensuremath{\\cat{Man}^{\\mathrm{c}}}_d$ can be viewed equivalently as the ``homotopy category of null bordisms'' by which we mean the undercategory $\\smash{h\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}_{\\varnothing\/}}$ of the empty manifold $\\varnothing$ viewed as an object in the homotopy category $h\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}$, whose objects are closed $(d-1)$-manifolds and whose morphisms are diffeomorphism classes of compact bordisms.\n\\end{rem}\n\nAs preparation to the proof of \\cref{thm:abstract-k-type-invariance}, we show that the values of the functor are invariant under certain surgeries.\n\n\\begin{lem}\\label{lem:surgery-invariance}Let $F$ be as in  \\cref{thm:abstract-k-type-invariance}. If two compact $d$-manifolds $M$ and $N$ differ by $p$-surgeries in the interior with $k\\le p\\le d-k-1$, then there exists an isomorphism $F(M)\\cong F(N)$.\n\\end{lem}\n\n\\begin{proof}\n\tIt suffices to show the claim in the case where $N$ is obtained from $M$ by a single $p$-surgery along an embedding $S^p\\times D^{d-p}\\hookrightarrow \\mathrm{int}(M)$. We consider the zig-zag \n\t\\[\n\t\tF(M)\\longleftarrow F(M\\backslash \\mathrm{int}(S^p\\times D^{d-p}))\\longrightarrow F(N)\n\t\\]\n\tinduced by the inclusions $M\\backslash \\mathrm{int}(S^p\\times D^{d-p})\\subset M$ and $M\\backslash \\mathrm{int}(S^p\\times D^{d-p}) \\subset N$. The former has relative handle type $[d-p,d]$ and the latter has relative handle type $[p+1,d]$. As $d-p\\ge k+1$ and $p+1\\ge k+1$ and $F$ sends inclusions of submanifolds of relative handle type $[k+1,d]$ to isomorphisms by assumption, we conclude the claim.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{thm:abstract-k-type-invariance}]Recall that two $d$-manifolds $M$ and $N$ have the same tangential $k$-type if there exists a map $\\theta\\colon B\\rightarrow \\mathrm{BO}$ and $k$-connected $\\theta$-structures $\\ell_M$ and $\\ell_N$ on $M$ and $N$. \n\t\nPart \\ref{k-type-i} of the claim asserts an isomorphism $F(M)\\cong F(N)$. In the case that $(M,\\ell_M)$ and $(N,\\ell_N)$ are closed manifolds that are $\\theta$-bordant, this follows directly from \\cref{cor:theta-bordism-surgery} and \\cref{lem:surgery-invariance}. To show the general case, we pick a handle decomposition of $M$ viewed as a bordism $M\\colon \\varnothing\\leadsto \\partial M$ and consider the zig-zag (using the notation from \\cref{sec:handle-dec})\n\\begin{equation}\\label{equ:reduction-to-closed-case}\n\tF\\big(M\\big)\\longleftarrow F\\big(M{[0,k]}\\big)\\longrightarrow F\\big(M[0,k]\\cup_{M_k}\\overline{M[0,k]}\\big)\n\\end{equation}\nwhose arrows are induced by the bordisms $M[k+1,d]\\colon M_k\\leadsto \\partial M$ and $\\overline{M{[0,k]}}\\colon M_k\\leadsto \\varnothing$. The former is of handle type $[k+1,d]$ and the latter of handle type $[d-k,d]$, so using that $d-k>k$ the two submanifold inclusions inducing the maps in the zig-zag have relative handle type $[k+1,d]$, so the zig-zag consists of isomorphisms. Applying the same reasoning for $N$, we see that the claim follows once we provide an isomorphism between the values of $F$ at the two doubles $M[0,k]\\cup_{M_k}\\overline{M[0,k]}$ and $N[0,k]\\cup_{N_k}\\overline{N[0,k]}$. Both of these doubles are closed manifolds that are $\\theta$-nullbordant (see \\cref{sec:theta-manifolds}), so they are in particular $\\theta$-bordant to each other. This implies the claim by the first part as long as we make sure that the induced $\\theta$-structures on these doubles are $k$-connected. But this is case, since it holds for $M$ and $N$ by assumptions and the above handle considerations in particular imply that all inclusions in $M\\supset M[0,k]\\subset M[0,k]\\cup_{M_k}\\overline{M[0,k]}$ and $N\\supset N[0,k]\\subset N[0,k]\\cup_{N_k}\\overline{N[0,k]}$ are $k$-connected. \n\t\nTo prove part~\\ref{k-type-ii}, we fix an embedding $L\\hookrightarrow M$ as in the claim which we may assume by transversality to be contained in $M[0,k]\\subset M$ as the complement $M[k+1,d]\\supset \\partial M$ has relative handle dimension $\\le d-(k+1)$ and $L$ has handle dimension $\\le k$ by assumption. The zig-zag \\eqref{equ:reduction-to-closed-case} is then compatible with the maps from $F(L)$ induced by inclusion. Now $M[0,k]\\cup_{M_k}\\overline{M[0,k]}$ differs from $N[0,k]\\cup_{N_k}\\overline{N[0,k]}$ by surgeries of index $k\\le p\\le d-k-1$, which we may assume (again by transversality) to be done away from $L$, so there is an embedding $L\\hookrightarrow N[0,k]\\cup_{N_k}\\overline{N[0,k]}$, such that the induced isomorphism $F(M[0,k]\\cup_{M_k}\\overline{M[0,k]})\\cong F(N[0,k]\\cup_{N_k}\\overline{N[0,k]})$ is compatible with the maps from $F(L)$. Using transversality on last time, we see that we may isotope the embedding $L\\hookrightarrow N[0,k]\\cup_{N_k}\\overline{N[0,k]}$ to land in $N[0,k]$ since $\\overline{N[0,k]}\\subset $ has handle dimension $\\le k$ and $2k< d$. With respect to the isotoped embedding, the zig-zag of equivalences \\eqref{equ:reduction-to-closed-case} is compatible with the maps from $F(N)$ and this concludes the proof.\n\t\nFor part~\\ref{k-type-iii}, we may assume without loss of generality that the embedding is a submanifold inclusion of the form $M\\subset M\\cup_{\\partial M} W$ for $W\\colon \\partial M\\leadsto \\partial N$ a bordism. We now consider the commutative square of codimension $0$ submanifold inclusions\n\\[\\begin{tikzcd}\n\tc(M_k)\\arrow[d,hook]\\arrow[r,hook]& c(M_k)\\cup_{M_k}M[k+1,d]\\cup_{\\partial M}W\\arrow[d,hook]\\\\\n\tM\\arrow[r,hook]&M\\cup_{\\partial M}W=N\n\\end{tikzcd}\\]\nwhere $c(M_k)\\subset M$ is a closed bicollar of $M_k\\subset M$. The vertical inclusions are of relative handle type $[k+1,d]$ (this uses $d-k\\ge k+1$), so we conclude that they map to isomorphisms under $F$. It thus suffices to show that $F$ maps the top horizontal inclusion to an isomorphism. Since the vertical inclusions and the $\\theta$-structures $\\ell_M$ and $\\ell_N$ are $k$-connected, it follows that the top horizontal inclusion induces an isomorphism on $\\pi_i(-)$ for $i\\le k-1$, so the same holds for the inclusion $M_k\\subset  c(M_k)\\cup_{M_k}M[k+1,d]\\cup_{\\partial M}W$. Following the argument in the proof of \\cref{lem:bordism-connectivity}, we may change $c(M_k)\\cup_{M_k}M[k+1,d]\\cup_{\\partial M}W$ by $k$-surgeries away from the collar $c(M_k)$ to a bordism $(c(M_k)\\cup_{M_k}V)\\colon M_k\\leadsto \\partial N$ such that $M_k\\subset (c(M_k)\\cup_{M_k} V)$ is $k$-connected. By \\cref{lem:surgery-invariance} and its proof we obtain an isomorphism $F(c(M_k)\\cup_{M_k}M[k+1,d]\\cup_{\\partial M}W)\\cong F(c(M_k)\\cup_{M_k} V)$ that is compatible with the maps from $F(c(M_k))$, so we are left with showing that the map $F(c(M_k))\\rightarrow F(c(M_k)\\cup_{M_k} V)$ is an isomorphism. But since $M_k\\subset V$ is $k$-connected and the assumption $2\\le k<\\tfrac{d-1}{2}$ implies $k\\le d-4$, the bordism $ V\\colon M_k\\leadsto \\partial N$ is of handle type $[k+1,d]$ by handle trading \\cite[Theorem 3]{WallConnectivity}, so $F(c(M_k))\\rightarrow F(c(M_k)\\cup_{M_k} V)$ is an isomorphism.\n\\end{proof}\n\n\\subsection{$2$-type invariance of the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space}\\label{sec:2-type-invariance-sdisc} \nBy \\cref{sec:sdisc-for-manifolds}, the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces of compact manifolds form the values of a functor $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(-)\\colon \\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}_{\\varnothing\/}\\rightarrow \\ensuremath{\\catsingle{S}}$ of $\\infty$-categories, which induces on homotopy categories in view of \\cref{rem:nulbbordism-cat} a functor\n\\[\n\tS^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(-)\\colon h\\ensuremath{\\cat{Man}^{\\mathrm{c}}}_d\\simeq  h\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}_{\\varnothing\/}\\longrightarrow h\\ensuremath{\\catsingle{S}}.\n\\]\n\nThe goal of this section is to show that this functor satisfies the assumptions of \\cref{thm:abstract-k-type-invariance} for $k=2$. This can be rephrased as follows:\n \n\\begin{prop}\\label{prop:invariance-handles}\n\tLet $M\\coloneqq \\varnothing \\leadsto P$ and $W\\colon P \\leadsto Q$ be $d$-dimensional bordisms. If $W$ is of handle type $[3,d]$, then the gluing map\n\t$(-\\cup_P W)\\colon S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)\\rightarrow S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M\\cup_PW)$\n\tis an equivalence. \n\\end{prop}\n\nOnce this is proved, \\cref{thm:abstract-k-type-invariance} implies the following refined version of \\cref{bigthm:2-type-invariance}.\n\n\\begin{thm}\\label{thm:2-type-invariance-detailed} Let $d\\ge5$, and $M$, $N$ be two compact $d$-manifolds of the same tangential $2$-type.\n\\begin{enumerate}\n\t\\item\\label{enum:2-type-i} There exists an equivalence $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)\\simeq S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(N)$.\n\t\\item\\label{enum:2-type-ii} For any embedding $e\\colon L\\hookrightarrow M$ of a $d$-manifold $L$ with handle dimension $\\le2$ there is an embedding $e'\\colon L\\hookrightarrow N$ so that the equivalence of \\ref{enum:2-type-i} can be chosen to be compatible with\n\t\\[\n\t\te_*\\colon S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(L)\\rightarrow S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)\\quad\\text{and}\\quad e'_*\\colon S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(L)\\rightarrow S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(N).\n\t\\]\n\t\\item\\label{enum:2-type-iii} If $d\\ge6$, then any embedding $e\\colon M\\hookrightarrow N$ that induces an equivalence on tangential $2$-types induces an equivalence $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)\\simeq S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(N)$ as in \\ref{enum:2-type-i}.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}[Proof of \\cref{prop:invariance-handles}]Unravelling the statement using \\cref{def:disc-structure-space}, the task is to show that\n\\[\\begin{tikzcd}\n\t\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P}\\dar{E}\\rar{(-)\\cup_PW}&[30pt] \\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{Q}\\dar{E}\\\\\n\t\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\mathrm{rep},\\simeq}_{E_{P \\times I}}\\rar{(-)\\cup_{E_{P \\times I}}E_W}& \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\mathrm{rep},\\simeq}_{E_{Q \\times I}}\n\\end{tikzcd}\\]\nis a pullback in $\\ensuremath{\\catsingle{S}}$, where $\\smash{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\mathrm{rep},\\simeq}_{E_{P \\times I}}\\subset \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{E_{P \\times I}}}$ and $\\smash{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\mathrm{rep},\\simeq}_{E_{Q \\times I}}\\subset \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{E_{Q \\times I}}}$ are the $\\infty$-groupoids given as the full subcategories of those objects in the image of the functor $E\\colon \\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P}\\rightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{E_{P \\times I}}$ and in the image of its analogue for $P$ replaced by $Q$, respectively. We prove that it is a pullback by showing that the map on vertical fibres are equivalences, for which we use that for any map $f\\colon E\\rightarrow B$ in $\\ensuremath{\\catsingle{S}}$ (thought of as a full subcategory of $\\ensuremath{\\icat{C}\\mathrm{at}_\\infty}$) and a point $b\\in B$, the fibre of $f$ over $b$ agrees with the colimit $\\mathrm{colim}_{E}\\,\\mathrm{Map}_B(f(-),b)$. This follows from \\cite[3.3.4.6]{LurieHTT} combined with the fact that the fibre over $b$ is the total space of the unstraightening of the functor $\\mathrm{Map}_B(f(-),b)\\colon E\\rightarrow \\ensuremath{\\catsingle{S}}$ which in turn follows from \\cite[3.3.2.8]{LurieHTT}. \n\nApplying this to the situation at hand and using the description of $E$ on mapping spaces from \\cref{sec:functor-e-summary}, the claim follows once we show that for each nullbordism $N\\in \\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{Q}$, the map\n\\vspace{-0.05cm}\n\\[\n\t\\underset{(\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P})^\\mathrm{op}}{\\mathrm{colim}}\\big[\\mathrm{Map}_{\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{Q}}\\big((-)\\cup_P W,N \\big)\\big]\\xlra{E} \\underset{(\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\mathrm{rep},\\simeq}_{E_{P \\times I}})^\\mathrm{op}}{\\mathrm{colim}}\\big[\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{E_{Q \\times I}}}\\big((-)\\cup_{E_{P \\times I}}E_W,E_N\\big)\\big]\n\\]\nis an equivalence. Using the factorisation $E\\colon \\ensuremath{\\icat{B}\\mathrm{ord}}(d)\\to \\ensuremath{\\icat{M}\\mathrm{od}}(d)$ through the noncompact version of the bordism double $\\infty$-category $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)$, this map fits into a commutative diagram\n\\[\\hspace{-0.3cm}\\begin{tikzcd}[column sep=0.2cm,row sep=0.5cm,ar symbol\/.style = {draw=none,\"\\textstyle#1\" description,sloped},\n\tequ\/.style = {ar symbol={\\simeq}}]\n\t\\underset{(\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P})^\\mathrm{op}}{\\mathrm{colim}}\\big[\\mathrm{Map}_{\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{Q}}\\big((-)\\cup_P W,N\\big)\\big]\\dar{\\circled{1}}\\rar{E}& \\underset{(\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\mathrm{rep},\\simeq}_{E_{P \\times I}})^\\mathrm{op}}{\\mathrm{colim}}\\big[\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{E_{Q \\times I}}}\\big((-)\\cup_{E_{P \\times I}}E_W,E_N\\big)\\big]\\dar{\\circled{2}}\\\\\n\t\\underset{(\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P})^\\mathrm{op}}{\\mathrm{colim}}\\big[\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{Q}}\\big((-)\\cup_P W,N\\big)\\big]\\rar{E} & \\underset{(\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\mathrm{rep}}_{E_{P \\times I}})^\\mathrm{op}}{\\mathrm{colim}}\\big[\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{Q}}\\big((-)\\cup_{E_P}E_W,E_N\\big)\\big] \\\\[-0.1cm]\n\t\\mathrm{Map}_{\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{Q}}\\big(P\\times(-1,0]\\cup_P W,N\\big)\\rar{E}\\arrow[u,\"\\simeq\",shorten >=-6pt] &\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q \\times I}}}\\big(E_{P\\times(-1,0]\\cup_P W},E_N\\big)\\arrow[u,\"\\simeq\",swap,shorten >=-8pt]\n\\end{tikzcd}\\]\nwhere the bottom equivalences result from the fact that the bordism $(P\\times (-1,0])\\colon \\varnothing\\leadsto P$ is initial in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P}$ and its image under $E$ is initial in $\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\mathrm{rep}}_{E_{P \\times I}}$, by \\cref{rem:initial-among-rep-presheaves}. By \\cref{cor:convergence} the bottom map is an equivalence as the handle dimension of $P\\times(-1,0]\\cup_P W$ relative to $Q$ is $\\le d-3$ by assumption. It thus suffices to show that $\\circled{1}$ and $\\circled{2}$ are equivalences.\n\nWe begin with $\\circled{1}$. Since the mapping spaces in $\\ensuremath{\\mathrm{nc}\\icat{B}\\mathrm{ord}}(d)_{P}$ are given by spaces of embeddings fixing the boundary and composition is given by composition of embeddings (see \\cref{sec:details-ncbord}) and the same holds for $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P}$ with embeddings replaced by diffeomorphisms (see \\cref{sec:details-bord}), the map $\\circled{1}$ is the map induced by restriction \n\\[\n\t\\underset{(\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P})^\\mathrm{op}}\\mathrm{colim} \\, \\ensuremath{\\mathrm{Diff}}_\\partial((-)\\cup_P W,N)\\longrightarrow \\ensuremath{\\mathrm{Emb}}_Q(W,N).\n\\]\nUsing the decomposition $\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P}=\\bigsqcup_{M\\in \\pi_0\\,\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P}}\\ensuremath{\\mathrm{BDiff}}_\\partial(M)$ into path components (see \\cref{sec:details-bord}), this can further be simplified as\n\\begin{equation}\\label{equ:reduction-of-circled-1}\\begin{tikzcd}[row sep=0.33cm] \n\t\\underset{M\\in \\pi_0\\,\\ensuremath{\\icat{B}\\mathrm{ord}}(d)_{P}}\\bigsqcup \\ensuremath{\\mathrm{Diff}}_\\partial(M\\cup_P W,N)\/ \\ensuremath{\\mathrm{Diff}}_\\partial(M)  \\dar[shorten <=-9pt] \\\\[-8pt] \n\t\\ensuremath{\\mathrm{Emb}}_Q(W,N).\n\\end{tikzcd}\\end{equation}\nTo show that the map \\eqref{equ:reduction-of-circled-1} is an equivalence, we show  separately that it induces a bijection on components and that it is an equivalence on each component. To see that it is surjective on components, pick an embedding $e\\in\\ensuremath{\\mathrm{Emb}}_Q(W,N)$. Up to changing $e$ within its isotopy class, we can assume that $P\\subset W$ is mapped to the interior of $N$ and that the complement of $e(W\\backslash P)\\subset N$ defines a bordism $(N\\backslash e(W\\backslash P))\\colon P\\leadsto Q$. In this case the class in $\\pi_0\\,\\ensuremath{\\mathrm{Diff}}_\\partial(M\\cup_P W,N)\/ \\pi_0\\,\\ensuremath{\\mathrm{Diff}}_\\partial(M)=\\pi_0(\\ensuremath{\\mathrm{Diff}}_\\partial(M\\cup_P W,N)\/ \\ensuremath{\\mathrm{Diff}}_\\partial(M))$ of the diffeomorphism $(N\\backslash e(W\\backslash P))\\cup_P W\\cong N$ obtained by extending $e$ by the identity provides a preimage of $[e]\\in\\pi_0\\,\\ensuremath{\\mathrm{Emb}}_Q(W,N)$. Injectivity of \\eqref{equ:reduction-of-circled-1} on $\\pi_0$ follows from the isotopy extension theorem in the form of the homotopy fibre sequence\n\\vspace{-0.2cm}\n\\[\n\t\\ensuremath{\\mathrm{Diff}}_\\partial(M)\\xrightarrow{\\phi\\circ ((-)\\cup_P{\\mathrm{id}_W})} \\ensuremath{\\mathrm{Diff}}_\\partial(M\\cup_PW,N)\\xra{\\mathrm{res}} \\ensuremath{\\mathrm{Emb}}_Q(W,N)\n\\]\nwith fibre taken over the image of a diffeomorphism $\\phi\\colon M\\cup_PW\\cong N$. This sequence also implies that \\eqref{equ:reduction-of-circled-1} is an equivalence on components, which finishes the proof for $\\circled{1}$.\n\nThe argument for $\\circled{2}$ is similar. Using Sections~\\ref{sec:details-bimod} and~\\ref{sec:functor-e-summary}, the reduction to showing that \\eqref{equ:reduction-of-circled-1} is an equivalence applies also to the map $\\circled{2}$ and shows that it agrees with the map\n\\[\\begin{tikzcd}[row sep=0.33cm] \n\t\\underset{E_M\\in \\pi_0\\,\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\mathrm{rep},\\simeq}_{E_{P \\times I}}}\\bigsqcup \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q \\times I}}^{\\simeq}}(E_{M\\cup_PW},E_N)\/ \\mathrm{Aut}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{E_{P \\times I}}}(E_M) \\dar[shorten <=-9pt] \\\\[-8pt] \n\t\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q \\times I}}^{\\simeq}}(E_{P\\times (-1,0]\\cup_PW},E_N);\n\\end{tikzcd}\\]\ninduced by the inclusion $P\\times (-1,0]\\cup_PW\\subset M\\cup_PW$. From the commutativity of the big diagram above and the fact that $\\circled{1}$ and the bottom horizontal map are equivalences, we see that $\\circled{2}$ is surjective on $\\pi_0(-)$, so we are left to show that it is injective on $\\pi_0(-)$ and that it induces an equivalence on components. This follows as for $\\circled{1}$ once we show that for $E_M\\in\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\mathrm{rep},\\simeq}_{E_{P \\times I}}$ and $\\phi\\in \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q \\times I}}^{\\simeq}}(E_M\\cup_{E_P} E_W,E_N)$ the sequence\n\\vspace{-0.2cm}\n\\[\n\t\\mathrm{Aut}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{E_{P \\times I}}}(E_M)\\longrightarrow \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{E_{Q \\times I}}}(E_{M\\cup_PW},E_{N})\\xra{\\mathrm{res}} \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{Q \\times I}}}(E_{P\\times(-1,0]\\cup_PW},E_{N}),\n\\]\nwhose left map is given by $\\phi\\circ ((-)\\cup_{E_{P \\times I}}{\\mathrm{id}_{E_W}})$, is a homotopy fibre sequence when taking homotopy fibres over the image of $\\phi$. By postcomposition with an inverse of $\\phi$ it suffices to show this in the case $\\phi=\\mathrm{id}$. This follows from the second part of \\cref{thm:isotopy-extension} (set $P=\\varnothing$, $Q=P$, $R=Q$, $W=\\varnothing$, $W'=W$, $M=M$, and $N=M$). The hypothesis to apply this result holds by \\cref{cor:convergence}, since it follows from the assumption that $\\underline{k}\\times\\ensuremath{\\mathbf{R}}^d\\sqcup P\\times(-1,0]\\cup_PW$ is the interior of a manifold obtained from a closed collar on $Q$ by attaching $(\\le d-3)$-handles for all $k$.\n\\end{proof}\n\n\nWe conclude this section with a first application of the tangential $2$-type invariance. We will later use it to reduce the proof of the nontriviality result for $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ to the case of $M=D^d$.\n\n\\begin{cor}\\label{cor:homotopy-retract}For a compact spin $d$-manifold $M\\neq\\varnothing$ with $d\\ge5$, the space $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ contains $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^d)$ as a homotopy retract. \n\\end{cor}\n\n\\begin{proof}This essentially follows from the fact that any finitely presented group arises as the fundamental group of a compact connected codimension $0$-submanifold $N\\subset D^k$ as long as $k\\ge 5$ (in fact $k \\geq 4$ is known to suffice, but we will not need this harder result). Indeed, apply this to $k=d$ and the fundamental group of each path component of $M$, to obtain a compact $d$-manifold $N\\subset D^d$ whose fundamental groupoid is equivalent to that of $M$. Since $N$ admits an embedding into $D^d$, it is in particular spin, so the final discussion in \\cref{rem:classification-2-types} shows that $M$ and $N$ have the same tangential $2$-type. Using the tangential $2$-type invariance of $S^{\\ensuremath{\\cat{Disc}}}_\\partial(-)$ from \\cref{thm:2-type-invariance-detailed}, it thus suffices to show the claim for $N$. The latter follows by choosing an embedded disc $D^d\\subset N$ so that the composition $D^d\\subset N\\subset D^d$ is isotopic to the identity and applying $S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(-)$.\n\\end{proof}\n\n\n\\section{\\cref{bigthm:infinite-loop-space}: infinite loop space} \\label{sec:infinite-loop-space} \nThe goal of this section is the proof of \\cref{bigthm:infinite-loop-space}, or rather the following strengthening of it:\n\n\\begin{thm}\\label{thm:oo-loop-general}For a compact manifold $M$ of dimension $d \\geq 8$, $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ admits the structure of an infinite loop space. If $M$ is $1$-connected spin, then the bound $d\\ge8$ can be improved to $d\\ge6$.\\end{thm}\n\nIn \\cref{sec:intr-infinite-loop-space}, we already gave an informal overview of the proof. We now make it precise.\n\n\\subsection{Operads with homological stability}\nThe proof of \\cref{thm:oo-loop-general} relies on work of Basterra--Bobkova--Ponto--Tillmann--Yaekel \\cite{BBPTY} on \\emph{operads with homological stability} which generalises earlier work of Tillmann \\cite{Tillmann}. We summarise their main result in this subsection.\n\n\\begin{rem}\n\\cite{BBPTY} is written in the setting of classical operads in topological spaces and algebras over them. To make it fit in our framework, we will rephrase their result in terms of (symmetric) $\\infty$-operads (see \\cref{sec:gen-infty-operads}). This translation is justified by the fact that there is an equivalence of $\\infty$-categories between the $\\infty$-category $\\ensuremath{{\\icat{O}\\mathrm{pd}_\\infty}}$ of $\\infty$-operads and the $\\infty$-category underlying the model category $\\ensuremath{{s\\icat{O}\\mathrm{p}}}$ of classical coloured operads in simplicial sets (see \\cite[p.\\,858]{ChuHaugsengHeuts}) which is in turn Quillen equivalent to that of classical coloured operads in topological spaces. These equivalences do not affect the induced operad in the homotopy category, and they extend to equivalences between categories of algebras since, in both cases, algebras over a coloured operad $\\ensuremath{\\catsingle{O}}$ in a symmetric monoidal category $\\ensuremath{\\catsingle{C}}$ are nothing but morphisms of operads from $\\ensuremath{\\catsingle{O}}$ to the operad underlying $\\ensuremath{\\catsingle{C}}$ (with colours the objects in $\\ensuremath{\\catsingle{C}}$).\n\\end{rem}\n\nLet $\\ensuremath{\\mathbf{N}}_0$ denote the set of non-negative integers. To state the main result of \\cite{BBPTY}, we consider \\emph{$\\ensuremath{\\mathbf{N}}_0$-graded $\\infty$-operads} by which we mean (symmetric) $\\infty$-operads $\\ensuremath{\\catsingle{P}}$, together with a map of $\\infty$-operads $\\ensuremath{\\catsingle{P}}^{\\otimes}\\rightarrow \\gls*{finn}$ to the $\\infty$-operad $\\ensuremath{\\cat{Fin}}^{\\ensuremath{\\mathbf{N}}_0}_\\ast$ that is induced (via the operadic nerve \\cite[2.1.1.27]{LurieHA}) by $\\ensuremath{\\mathbf{N}}_0$ under addition, considered as a symmetric monoidal category with a single object. Unpacking the definition, this amounts to an $\\ensuremath{\\mathbf{N}}_0$-indexed disjoint union decomposition $\\mathrm{Mul}_\\ensuremath{\\catsingle{P}}(x_1,\\ldots,x_n;y)=\\sqcup_{g\\ge0}\\mathrm{Mul}_\\ensuremath{\\catsingle{P}}(x_1,\\ldots,x_n;y)_g$ of all spaces of multi-operations that is additive under operadic composition. Every $\\infty$-operad $\\ensuremath{\\catsingle{O}}$ can be viewed as an $\\ensuremath{\\mathbf{N}}_0$-graded operad in grading $0$; formally this amounts to considering the composition $\\ensuremath{\\catsingle{O}}^{\\otimes}\\rightarrow\\ensuremath{\\cat{Fin}}_\\ast\\rightarrow \\ensuremath{\\cat{Fin}}^{\\ensuremath{\\mathbf{N}}_0}_\\ast$ where the second arrow is induced by the inclusion $\\{0\\}\\subset \\ensuremath{\\mathbf{N}}_0$.\n\n\\begin{dfn}\\label{dfn:operad-w-homstab}An \\emph{operad with homological stability} is an $\\ensuremath{\\mathbf{N}}_0$-graded $\\infty$-operad $\\ensuremath{\\catsingle{P}}$ with a single colour (whose space of $k$-ary operations we write as $\\mathrm{Mul}_{\\ensuremath{\\catsingle{P}}}(\\ast,\\ldots,\\ast;\\ast)=\\ensuremath{\\catsingle{P}}(k)=\\sqcup_{g\\ge0}\\ensuremath{\\catsingle{P}}_g(k)$), together with\n\\begin{enumerate}\n\t\\item\\label{enum:data-homstab-operad-i} a map of $\\ensuremath{\\mathbf{N}}_0$-graded $\\infty$-operads $\\ensuremath{\\icat{A}\\mathrm{ssoc}} \\rightarrow \\ensuremath{\\catsingle{P}}$ from the associative operad $\\ensuremath{\\icat{A}\\mathrm{ssoc}}$ (see \\cref{ex:associative-operad}) concentrated in degree $0$, and\n\t\\item\\label{enum:data-homstab-operad-ii} a distinguished element $s\\in \\ensuremath{\\catsingle{P}}_1(1)$, called the \\emph{stabilising element},\n\\end{enumerate}\nsuch that\n\\begin{enumerate}[(a)]\n\t\\item\\label{enum:cond-homstab-operad-i} the map on $2$-ary operations $\\ensuremath{\\icat{A}\\mathrm{ssoc}}(2)\\rightarrow \\ensuremath{\\catsingle{P}}_0(2)$ lands in a single path component, and\n\t\\item\\label{enum:cond-homstab-operad-ii} the map $\\ensuremath{\\catsingle{P}}_\\infty(k)\\coloneqq \\mathrm{colim}_g\\,\\ensuremath{\\catsingle{P}}_g(k)\\rightarrow\\mathrm{colim}_g\\,\\ensuremath{\\catsingle{P}}_g(0)\\eqcolon\\ensuremath{\\catsingle{P}}_\\infty(0)$ induced by taking horizontal colimits in the commutative diagram in $\\ensuremath{\\catsingle{S}}$\n\t\\[\\begin{tikzcd}\n\t\t\\cdots\\rar&\\ensuremath{\\catsingle{P}}_{g-1}(k)\\rar{\\circ_P(s;-)} \\arrow[d,\"{\\circ_P(-;*,\\ldots,*)}\"]&[10pt]\\ensuremath{\\catsingle{P}}_g(k)\\rar{\\circ_P(s;-)} \\arrow[d,\"{\\circ_P(-;*,\\ldots,*)}\"]&[10pt]\\ensuremath{\\catsingle{P}}_{g+1}(k)\\dar{\\circ_P(-;*,\\ldots,*)}\\rar&\\cdots\\\\\n\t\t\\cdots\\rar&\\ensuremath{\\catsingle{P}}_{g-1}(0)\\rar{\\circ_P(s;-)}&\\ensuremath{\\catsingle{P}}_g(0)\\rar{\\circ_P(s;-)}\\rar&\\ensuremath{\\catsingle{P}}_{g+1}(0)\\rar&\\cdots\n\t\\end{tikzcd}\\]\n\tis an integral homology isomorphism for all $k\\ge0$; here $\\circ_P(-;-)$ is the operadic composition and $*\\in \\ensuremath{\\catsingle{P}}_0(0)$ is the image of $\\ast \\simeq\\ensuremath{\\icat{A}\\mathrm{ssoc}}(0) \\rightarrow \\ensuremath{\\catsingle{P}}_0(0)$.\n\\end{enumerate}\n\\end{dfn}\n\nGiven $\\ensuremath{\\catsingle{P}}$ as in \\cref{dfn:operad-w-homstab}, we may forget the grading and consider the composition \n\\begin{equation}\\label{equ:forgetfulgroup-completion-functor}\n\\ensuremath{\\mathrm{Alg}}_\\ensuremath{\\catsingle{P}}(\\ensuremath{\\catsingle{S}}) \\longrightarrow \\ensuremath{\\mathrm{Alg}}_{\\ensuremath{\\icat{A}\\mathrm{ssoc}}}(\\ensuremath{\\catsingle{S}})\\underset{\\simeq}{\\xrightarrow{\\text{\\cite[p\\,465]{LurieHA}}}} \\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{S}})\\xlra{\\Omega B} \\ensuremath{\\mathrm{Mon}}^\\ensuremath{{\\mathrm{grp}}}(\\ensuremath{\\catsingle{S}})\\xlra{U}\\ensuremath{\\catsingle{S}}\n\\end{equation}\nwhere the first arrow the functor between $\\infty$-categories of algebras in $\\ensuremath{\\catsingle{S}}$ with its cartesian symmetric monoidal structure, induced by the morphism $\\ensuremath{\\icat{A}\\mathrm{ssoc}} \\rightarrow \\ensuremath{\\catsingle{P}}$ of $\\infty$-operads (see \\cref{sec:gen-infty-operads}), the second arrow is given by \\emph{group-completion}, i.e.\\,the left adjoint of the full subcategory inclusion $\\ensuremath{\\mathrm{Mon}}^\\ensuremath{{\\mathrm{grp}}}(\\ensuremath{\\catsingle{S}})\\subset \\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{S}})$ of \\emph{group-like objects}, i.e.\\,those monoid objects $M\\in\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{S}})\\subset \\ensuremath{\\mathrm{Fun}}(\\Delta^{\\mathrm{op}},\\ensuremath{\\catsingle{S}})$ in the sense of \\cref{sec:cat-objects} for which the induced monoid of path components $\\pi_0(M_{[1]})$ is a group, and the final arrow is the forgetful functor, given by evaluation at $[1]\\in\\Delta$. Recall (see e.g.\\cite[5.2.6]{LurieHA}) that the composition of the final two arrows sends $M\\in\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{S}})$ to the pullback in $\\ensuremath{\\catsingle{S}}$\n\\[\\begin{tikzcd} \n\t\\Omega BM \\rar \\dar & M_{[0]} \\simeq \\ast \\dar \\\\\n\t\\ast\\simeq M_{[0]} \\rar & BM,\n\\end{tikzcd} \n\\quad \\text{with} \\quad \nBM = \\underset{\\Delta^\\mathrm{op}}\\mathrm{colim}\\,M.\n\\]\nWriting $\\ensuremath{\\mathrm{Alg}}^\\ensuremath{{\\mathrm{grp}}}_{E_\\infty}(\\ensuremath{\\catsingle{S}})\\subset \\ensuremath{\\mathrm{Alg}}_{E_\\infty}(\\ensuremath{\\catsingle{S}})$ for the full subcategory of group-like algebras in $\\ensuremath{\\catsingle{S}}$ over the $E_\\infty$-operad \\cite[5.1.1.6]{LurieHA}, the main result of \\cite{BBPTY} reads as follows:\n\n\\begin{thm}[Basterra--Bobkova--Ponto--Tillmann--Yeakel]\\label{thm:stability-operads}\n\tFor an operad with homological stability $\\ensuremath{\\catsingle{P}}$, there exists a dashed functor fitting  into a commutative diagram of $\\infty$-categories\n\t\\[\\begin{tikzcd}\n\t\t&\\ensuremath{\\mathrm{Alg}}^\\ensuremath{{\\mathrm{grp}}}_{E_\\infty}(\\ensuremath{\\catsingle{S}})\\dar{U}\\\\\n\t\t\\ensuremath{\\mathrm{Alg}}_{\\ensuremath{\\catsingle{P}}}(\\ensuremath{\\catsingle{S}})\\rar{\\eqref{equ:forgetfulgroup-completion-functor}}\\arrow[ur,dashed]&\\ensuremath{\\catsingle{S}}\n\t\\end{tikzcd}\\]\n\twhere $U$ is the forgetful functor.\n\\end{thm}\nIn other words, the underlying space of the group completion of an algebra over an operad with homological stability (considered as an ungraded operad) admits functorially the structure of a group-like $E_\\infty$-algebra, or equivalently---by the recognition principle \\cite[5.2.6.26]{LurieHA}---that of an infinite loop space.\n\n\\subsection{A manifold operad with homological stability}\nThe main example of an operad with homological stability considered in  \\cite{BBPTY} is constructed out of the manifolds\n\\[\n\tW^{2n}_{g,k+l} \\coloneqq W^{2n}_{0,k+l} \\sharp (S^n \\times S^n)^{\\sharp g}\\quad\\text{with}\\quad W^{2n}_{0,k+l}\\coloneqq S^{2n}\\backslash\\mathrm{int}((\\sqcup^{k}D^{2n})\\sqcup (\\sqcup^{l}D^{2n}))\n\\]\nfor $k,l\\ge0$ and $n\\ge1$, considered as bordisms of the form $\\sqcup^kS^{2n-1}\\leadsto \\sqcup^lS^{2n-1}$. Here $\\smash{\\sharp}$ denotes the connected sum operation. This is also the operad that is relevant for the proof of \\cref{thm:oo-loop-general}, so we recall its construction in our setting. We omit the $2n$-superscripts for brevity.\n\nConsider the tangential structure $\\theta=\\tau^*\\mathrm{Fr}(\\gamma)$ in the sense of \\cref{sec:details-tangential-bord} given as the $\\mathrm{GL}_{2n}(\\ensuremath{\\mathbf{R}})$-space which is the pullback of the frame bundle of the universal bundle $\\gamma\\rightarrow\\mathrm{BO}(2n)$ along the $n$-connected cover map $\\tau\\colon \\tau_{> n}\\mathrm{BO}(2n)\\rightarrow \\mathrm{BO}(2n)$. Since $S^{2n-1}$ is stably parallelisable, its once-stabilised tangent bundle admits a $\\theta$-structure $\\ell_0$ compatible its canonical orientation, unique up to equivalence of $\\theta$-structures. We consider the symmetric monoidal $\\infty$-category $\\ensuremath{\\icat{B}\\mathrm{ord}}^{\\theta}(2n)^{(\\infty,1)}$ from \\cref{sec:details-tangential-bord} and write $\\ensuremath{\\icat{B}\\mathrm{ord}}^{\\theta}(2n)^{(\\infty,1),W}$ for the sub symmetric monoidal $\\infty$-category (see \\cref{ex:sub-sym-monoidal}) obtained by restricting objects to those equivalent to $\\sqcup^k(S^{2n-1},\\ell_0)$ for $k\\ge0$ and restricting morphisms to those $\\theta$-bordisms whose underlying bordism without $\\theta$-structure is equivalent to a disjoint union of $W_{g,k+1}$'s for some $g,k\\ge0$. Up to issues with components and different models, \\cite[Theorem 1.3]{BBPTY} shows that the endomorphism operad\n\\[\n\t\\ensuremath{\\catsingle{W}}\\coloneq \\mathrm{End}_{\\ensuremath{\\icat{B}\\mathrm{ord}}^{\\theta}(2n)^{(\\infty,1),W}}(S^{2n-1},\\ell_0)\n\\]\nof $(S^{2n-1},\\ell_0)$ in this category (see \\cref{sec:end-operads}) can be enhanced to an operad with homological stability for all $2n\\ge2$. For completeness and to deal with these issues, we give a proof in our setting by adapting their argument. As in \\cite{BBPTY}, the main ingredient is a stable homological stability result of Galatius--Randal-Williams \\cite{GRWII} (for the case $2n=2$ one can use \\cite{Harer}). \n\n\\begin{prop}\\label{prop:manifold-operad-w-homstab}$\\ensuremath{\\catsingle{W}}$ admits the structure of an operad with homological stability for all $2n\\ge2$.\\end{prop}\n\n\\begin{proof} By definition and \\eqref{equ:mapping-cat-theta-bord-compact}, the space of $k$-ary operations\n\\[\n\t\\ensuremath{\\catsingle{W}}(k)=\\mathrm{Map}_{\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(2n)^{(\\infty,1),W}}\\big({\\sqcup^k} (S^{2n-1},\\ell_0),(S^{2n-1},\\ell_0)\\big)\n\\] \nis the $\\infty$-groupoid of $\\theta$-bordisms $\\sqcup^k (S^{2n-1},\\ell_0)\\leadsto (S^{2n-1},\\ell_0)$ that are, after forgetting $\\theta$-structures, equivalent to $W_{g,k+1}$ for some $g \\geq 0$. As the manifolds $W_{g,k+1}$ are pairwise non-diffeomorphic for $g\\ge0$, this induces a decomposition $\\ensuremath{\\catsingle{W}}(k)=\\sqcup_{g\\ge0}\\ensuremath{\\catsingle{W}}(k)_g$ which is compatible with the operad structure given by gluing bordisms with $\\theta$-structures, so it gives rise to an $\\ensuremath{\\mathbf{N}}_0$-grading on $\\ensuremath{\\catsingle{W}}$.\n\nTo construct a map $\\ensuremath{\\icat{A}\\mathrm{ssoc}}\\rightarrow \\ensuremath{\\catsingle{W}}$ from the associative operad (put in degree $0$), we first use \\cref{ex:associative-operad} to recognise $\\ensuremath{\\icat{A}\\mathrm{ssoc}}$ as a sub-operad in the sense of \\cref{sec:suboperad} of the endomorphism operad $\\mathrm{End}_{\\ensuremath{\\icat{B}\\mathrm{ord}}^{\\mathrm{fr}}(2)^{\\partial,(\\infty,1)}}(D^1,\\mathrm{st})$ of the $1$-disc with the standard $1$-framing (that is, framing of its once-stabilised tangent bundle) considered as an object of the $2$-dimensional framed bordism category with boundary from \\cref{sec:details-tangential-bord} (formally, the tangential structure involved is $\\mathrm{fr}=(\\mathrm{id}\\colon \\mathrm{GL}_{2}(\\ensuremath{\\mathbf{R}})\\rightarrow \\mathrm{GL}_{2}(\\ensuremath{\\mathbf{R}})$). Namely, we restrict to those bordisms $(N,\\ell)\\colon {\\sqcup^k}(D^1,\\mathrm{st})\\leadsto (D^1,\\mathrm{st})$ for which $(N,\\ell)$ is diffeomorphic (after smoothing corners) to $D^2$ with its standard framing such that ${\\sqcup^k}(D^1,\\mathrm{st})\\subset \\partial D^2$ is orientation preserving, $(D^1,\\mathrm{st})\\subset \\partial D^2$ is orientation-reversing (see \\cref{fig:bord-int} for an example). From \\cref{ex:associative-operad}, one sees that this suboperad is equivalent to $\\ensuremath{\\icat{A}\\mathrm{ssoc}}$ since its space of $k$-ary operations is homotopy discrete with components $\\Sigma_k$ (with the regular $\\Sigma_k$-action) as a consequence of the facts that (i) the diffeomorphism group of $D^2$ fixing some boundary intervals is contractible as a result of the equivalences $\\ensuremath{\\mathrm{Diff}}_\\partial(D^1)\\simeq\\ast$ and $\\ensuremath{\\mathrm{Diff}}_\\partial(D^2)\\simeq\\ast$ (the first is folklore, the latter is \\cite[Theorem B]{Smale}) and that (ii) the space of framings of $D^2$ relative to fixed $1$-framings on collared intervals in the boundary is homotopy discrete (as $\\Omega \\mathrm{GL}_2(\\ensuremath{\\mathbf{R}})$ is).\n\nNow we consider the composition of symmetric monoidal $\\infty$-categories\n\\vspace{-0.1cm}\n\\[\n\t\\ensuremath{\\icat{B}\\mathrm{ord}}^{\\mathrm{fr}}(2)^{\\partial,(\\infty,1)}\\xra{(-)\\times(D^{2n-1},\\mathrm{st})} \\ensuremath{\\icat{B}\\mathrm{ord}}^{\\mathrm{fr}}(2n+1)^{\\partial,(\\infty,1)}\\xlra{\\partial} \\ensuremath{\\icat{B}\\mathrm{ord}}^{\\mathrm{1{-}fr}}(2n)^{(\\infty,1)}\\longrightarrow\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(2n)^{(\\infty,1)}\n\\]\nwhere the first arrow takes the product with $D^{2n-1}$ equipped with the standard framing and smooths corners (see \\cref{ex:product-functor-with-framing}), the second arrow takes boundaries and lands in the bordism category with $1$-stabilised framings (see \\cref{ex:framing-to-oneframing}), and the final arrow is induced by the naturality \\eqref{eqn:bord-theta-naturality} in the tangential structure and the fact that there is a map of tangential structures $(\\mathrm{1{-}fr}) \\to \\theta$ since $\\tau\\colon \\tau_{>n}\\mathrm{BO}(2n)\\rightarrow\\mathrm{BO}(2n)$ arises as the pullback of $\\tau_{>n}\\mathrm{BO}(2n+1)\\rightarrow \\mathrm{BO}(2n+1)$ along $\\mathrm{BO}(2n)\\rightarrow\\mathrm{BO}(2n+1)$ and thus receives a map from the pullback of $* \\to \\mathrm{BO}(2n+1)$. Taking endomorphism operads and precomposing with the map from $\\ensuremath{\\icat{A}\\mathrm{ssoc}}$, we have a composition \\vspace{-0.1cm} \n\\[\n\t\\ensuremath{\\icat{A}\\mathrm{ssoc}} \\overset{\\subset}\\longrightarrow \\mathrm{End}_{\\ensuremath{\\icat{B}\\mathrm{ord}}^{\\mathrm{fr}}(2)^{\\partial,(\\infty,1)}}(D^1,\\mathrm{st})\\longrightarrow\\mathrm{End}_{\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(2n)^{(\\infty,1)}}(S^{2n-1},\\ell_0)\n\\]\nwhich lands in the suboperad of $\\smash{\\mathrm{End}_{\\ensuremath{\\icat{B}\\mathrm{ord}}^{\\theta}(2n)^{(\\infty,1)}}(S^{2n-1},\\ell_0)}$ whose underlying bordisms are equivalent to $W_{0,k+1}$, using that $\\partial(D^2\\times D^{2n-1})\\backslash \\mathrm{int}(\\sqcup^{k+1}D^1\\times D^{2n-1})\\cong W_{0,k+1}$ after smoothing corners. In other words, it lands in the degree $0$-part of the operad $\\ensuremath{\\catsingle{W}}$ and thus gives a map $\\ensuremath{\\icat{A}\\mathrm{ssoc}} \\rightarrow \\ensuremath{\\catsingle{W}}$ as in part \\ref{enum:data-homstab-operad-i} of \\cref{dfn:operad-w-homstab}. As $s\\in \\ensuremath{\\catsingle{W}}_1(1)$ in part \\ref{enum:data-homstab-operad-ii}, we choose the bordism $W_{1,1}\\colon S^{2n-1}\\leadsto S^{2n-1}$ with an admissible $\\theta$-structure as in \\cite[p.\\,130]{GRWII} that extends $\\ell_0$ on the boundary spheres. \n\nThis leaves us with checking conditions \\ref{enum:cond-homstab-operad-i} and \\ref{enum:cond-homstab-operad-ii} of \\cref{dfn:operad-w-homstab}. For \\ref{enum:cond-homstab-operad-i}, one observes that already the composition $\\smash{\\ensuremath{\\icat{A}\\mathrm{ssoc}}(2)\\rightarrow\\mathrm{End}_{\\ensuremath{\\icat{B}\\mathrm{ord}}^\\mathrm{fr}(2)^{\\partial,(\\infty,1)}}(D^1,\\mathrm{st})(2)\\longrightarrow \\mathrm{End}_{\\ensuremath{\\icat{B}\\mathrm{ord}}^\\mathrm{fr}(2n+1)^{\\partial,(\\infty,1)}}(D^{2n},\\mathrm{st})(2)}$\nlands in a single path component, since the bordism $(D^{2n+1},\\mathrm{st})\\colon{\\sqcup^2}(D^{2n},\\mathrm{st})\\leadsto (D^{2n},\\mathrm{st})$ is for $n\\ge1$ framed diffeomorphic to the same bordism with the two source components permuted as consequence of the isotopy extension theorem and the fact that the space of framed embeddings $\\sqcup^2 D^{d}\\hookrightarrow D^{d}$ is connected for $d\\ge2$.\n  \nFinally, to verify \\ref{enum:cond-homstab-operad-ii} we note that the image of $\\ast\\simeq\\ensuremath{\\icat{A}\\mathrm{ssoc}}(0)\\rightarrow\\ensuremath{\\catsingle{W}}$ is the bordism $D^{2n}\\colon S^{2n-1}\\leadsto \\varnothing$, equipped with some $\\theta$-structure, so the map $\\ensuremath{\\catsingle{W}}_\\infty(k)\\rightarrow \\ensuremath{\\catsingle{W}}_{\\infty}(0)$ is a homology equivalence as a result of applying \\cite[Theorem 1.3]{GRWII} to the bordism (with some $\\theta$-structure) \n\\[\n\t(D^{2n})^{\\sqcup k}\\sqcup (S^{2n-1}\\times [0,1])\\colon (S^{2n-1})^{\\sqcup k}\\sqcup S^{2n-1}\\leadsto S^{2n-1},\n\\]\nwhich, being $(n-1)$-connected relative to its source, satisfies the condition of that theorem.\n\\end{proof}\n\n\\begin{figure}\n\t\\begin{tikzpicture}[scale=.9]\n\t\t\\foreach \\i in {1,...,4} \n\t\t{\n\t\t\t\\draw[->,thick,Mahogany] (0,{.2+\\i}) -- (0,{.8+\\i});\n\t\t\t\\draw (0,{-.2+\\i}) to[out=0,in=0,looseness=2](0,{.2+\\i});\n\t\t}\n\t\t\\draw[->,thick,Mahogany] (0,.2) -- (0,.8);\n\t\t\\draw[->,thick,Mahogany] (3,{.2+2}) -- (3,{.8+2});\n\t\t\\draw (0,0.2) to[out=0,in=180] (3,2.2);\n\t\t\\draw (0,4.8) to[out=0,in=180] (3,2.8);\n\t\\end{tikzpicture}\n\t\\caption{A $5$-ary operation in the $\\infty$-operad $\\ensuremath{\\catsingle{W}}$.}\n\t\\label{fig:bord-int}\n\\end{figure}\n\n\\subsection{Group completion and $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure spaces}\\label{sec:map-of-w-algebras}\nFixing numbers $2\\le 2n\\le d$ and a closed $(d-2n)$-manifold $P$, we consider the sequence of symmetric monoidal $\\infty$-categories\n\\begin{equation}\\label{equ:theta-bord-to-morita}\n\t\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(2n)^{(\\infty,1),W}\\subset \\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(2n)^{(\\infty,1)}\\xrightarrow{U}\\ensuremath{\\icat{B}\\mathrm{ord}}(2n)^{(\\infty,1)} \\overset{P\\times -}\\longrightarrow \\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)} \\overset{E}\\longrightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1)}\n\\end{equation}\nwhere $U$ forgets tangential structures, $P\\times(-)$ takes products (see \\ref{step:product} in \\cref{sec:the-functor}), and the final functor is discussed in \\cref{sec:functor-e-summary}. \\eqref{equ:theta-bord-to-morita} lands in the sub symmetric monoidal $\\infty$-category \n\\[\n\t\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1),W}\\subset \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1)},\n\\] \nwhich is obtained by the restricting the objects to those equivalent to $E_{\\sqcup^k P\\times S^{2n-1}\\times I}$ for $k\\ge0$ and the morphisms to those bimodules equivalent to $E_{\\sqcup^m P\\times W_{g,k+1}}$ for some $m,k,g\\ge0$. We write \n\\[\n\t\\ensuremath{\\icat{B}\\mathrm{ord}}(2n)^{(\\infty,1),\\overline{W}}\\subset \\ensuremath{\\icat{B}\\mathrm{ord}}(2n)^{(\\infty,1)}\n\\] \nfor the symmetric monoidal sub $\\infty$-category obtained by restricting objects and morphisms to those that land in $\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1),W}\\subset \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1)}$. Taking endomorphism operads, we obtain a composition of maps of $\\infty$-operads\n\\[\\hspace{-0.4cm}\\ensuremath{\\icat{A}\\mathrm{ssoc}} \\rightarrow \\ensuremath{\\catsingle{W}}=\\mathrm{End}_{\\ensuremath{\\icat{B}\\mathrm{ord}}^\\theta(2n)^{(\\infty,1),W}}(S^{2n-1},\\ell_0) \\rightarrow \\mathrm{End}_{\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1),\\overline{W}}}(P\\times S^{2n-1})\\rightarrow \\mathrm{End}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1),W}}(P\\times S^{2n-1}). \n\\]\nOn $0$-ary operations, this in particular induces a map of $\\ensuremath{\\catsingle{W}}$-algebras (see \\cref{sec:map-as-algebra})\n\\begin{equation}\\label{equ:map-of-endo-monoids}\n\t\\mathrm{Map}_{\\ensuremath{\\icat{B}\\mathrm{ord}}(2n)^{(\\infty,1),\\overline{W}}}(\\varnothing, P\\times S^{2n-1})\\longrightarrow \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1),W}}(E_\\varnothing, E_{P\\times S^{2n-1}\\times I})\n\\end{equation}\nwhich we can also view as a map of $\\ensuremath{\\icat{A}\\mathrm{ssoc}}$-algebras in $\\ensuremath{\\catsingle{S}}$, or equivalently, one of monoid objects in $\\ensuremath{\\catsingle{S}}$. Going through the construction, the unit in $\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1),W}}(\\varnothing, E_{P\\times S^{2n-1}\\times I})$ is given by the bimodule $E_{P\\times D^{2n}}$ and the fibre at that object of \\eqref{equ:map-of-endo-monoids}, viewed as a map in $\\ensuremath{\\catsingle{S}}$, is exactly $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(P\\times D^{2n})$ from \\cref{sec:sdisc-for-manifolds}. Since the forgetful functor $\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{S}})\\rightarrow \\ensuremath{\\catsingle{S}}$ preserves limits, $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(P\\times D^{2n})$ inherits a monoid structure which fits into a pullback diagram in $\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{S}})$\n\\begin{equation}\\label{equ:sdisc-as-pullback}\n\t\\begin{tikzcd} S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(P \\times D^{2n}) \\rar \\dar & \\mathrm{Map}_{\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1),\\overline{W}}}(\\varnothing,P \\times S^{2n-1}) \\dar \\\\[-2pt]\n\t\\ast \\rar{E_{P \\times D^{2n}}} & \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1),W}}(E_\\varnothing,E_{P \\times S^{2n-1} \\times I}).\\end{tikzcd}\n\\end{equation}\t\nUnder mild conditions, this square remains a pullback after group-completion. We show this as the first part of the following proposition.\n\n\\begin{prop}\\label{prop:group-completion}Fix $2\\le 2n\\le d$ with $d\\ge6$ and a closed $(d-2n)$-manifold $P$.\n\\begin{enumerate}\n\t\\item \\label{enum:group-completion-i}If also $2n\\ge4$, then the pullback \\eqref{equ:sdisc-as-pullback} in $\\ensuremath{\\mathrm{Mon}}(\\ensuremath{\\catsingle{S}})$ remains a pullback after group-completion.\n\t\\item \\label{enum:group-completion-ii}$S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(P\\times D^{2n})$ is group-like when considered as a monoid object in $\\ensuremath{\\catsingle{S}}$. \n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nThe first part is an application of the following fact, which can be deduced from \\cite[Theorem 2.11]{Steimle}: if a map $\\varphi\\colon X\\rightarrow Y$ of monoid objects in $\\ensuremath{\\catsingle{S}}$ has the property that for all $y\\in Y$ there is an $x\\in X$ such that $\\varphi(x)=y$ and the following squares are pullbacks in $\\ensuremath{\\catsingle{S}}$\n\\[\\begin{tikzcd}\n\tX\\dar\\rar{(-)\\cdot x}&[10pt]X\\dar\\\\[-2pt]\n\tY\\rar{(-)\\cdot y}&Y\n\\end{tikzcd}\n\\quad \\text{and}\\quad\n\\begin{tikzcd}\n\tX\\dar\\rar{ x\\cdot (-)}&[10pt]X\\dar\\\\[-2pt]\n\tY\\rar{y\\cdot(-)}&Y,\n\\end{tikzcd}\\]\nthen group completion preserves pullbacks of monoid objects in $\\ensuremath{\\catsingle{S}}$ along the map $\\varphi\\colon X\\rightarrow Y$.\n\nTo conclude \\ref{enum:group-completion-i}, it thus suffices to check the condition for the right vertical map in \\eqref{equ:sdisc-as-pullback} which amounts to showing that the square in $\\ensuremath{\\catsingle{S}}$\n\\[\n\\begin{tikzcd}[column sep=2cm]\n\\mathrm{Map}_{\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1),\\overline{W}}}(\\varnothing,P \\times S^{2n-1}) \\rar{(-)\\Ydown (P\\times W_{g,1})} \\dar & \\mathrm{Map}_{\\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1),\\overline{W}}}(\\varnothing,P \\times S^{2n-1}) \\dar \\\\\n\t\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1),W}}(E_\\varnothing,E_{P \\times S^{2n-1} \\times I})\\rar{(-)\\Ydown E_{P\\times W_{g,1}}} & \\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1),W}}(E_\\varnothing,E_{P \\times S^{2n-1} \\times I})\n\\end{tikzcd}\n\\]\nis cartesian for all $g\\ge0$, where $(-)\\gls*{ppproduct}(-)$ denotes the monoid structure of the monoid objects in \\eqref{equ:sdisc-as-pullback}, and that the same holds for the square where we take products from the left. We focus on the former; the latter is proved in the same way.\n\nGoing through the construction of the map $\\ensuremath{\\icat{A}\\mathrm{ssoc}}\\rightarrow\\ensuremath{\\catsingle{W}}$ in the proof of \\cref{prop:manifold-operad-w-homstab}, we see that $(-)\\Ydown (P\\times W_{g,1})$ is given a ``pair of pants-product'': it sends a bordism $M\\colon \\varnothing \\leadsto P\\times S^{2n-1}$ to the disjoint union $(M\\sqcup P\\times W_{g,1})\\colon \\varnothing \\leadsto \\sqcup^2P\\times S^{2n-1}$ and then takes composition with $(P\\times W_{0,2+1})\\colon \\sqcup^2P\\times S^{2n-1}\\leadsto P\\times S^{2n-1}$. By monoidality, this agrees with the map that sends $M\\colon \\varnothing\\leadsto  P\\times S^{2n-1}$ first to its composition with $([0,1]\\times P\\times S^{2n-1}\\sqcup P\\times W_{g,1})\\colon P\\times S^{2n-1}\\leadsto \\sqcup^2 P\\times S^{2n-1}$ and then takes composition with $P\\times W_{0,2+1}\\colon \\sqcup^2P\\times S^{2n-1}\\leadsto P\\times S^{2n-1}$. The composition of the latter two bordisms is diffeomorphic, as a self-bordism of $P\\times S^{2n-1}$, to $ P\\times W_{g,1+1}$. The same argument applies to $(-)\\Ydown E_{P\\times W_{g,1}}$, so using monoidality of the functor $E\\colon  \\ensuremath{\\icat{B}\\mathrm{ord}}(d)^{(\\infty,1)}\\rightarrow \\ensuremath{\\icat{M}\\mathrm{od}}(d)^{(\\infty,1)}$, we may replace the top and bottom maps in the previous square by the gluing maps $(-)\\cup_{P\\times S^{2n-1}}(P\\times W_{g,1+1})$ and $(-)\\cup_{E_{P\\times S^{2n-1}\\times I}}E_{P\\times W_{g,1+1}}$ respectively. Taking vertical homotopy fibres, it thus suffices to show that for $h\\ge0$ the gluing map\n\\[\\big((-)\\cup_{P\\times S^{2n-1}}(P\\times W_{g,1+1})\\big)\\colon S_\\partial(P\\times W_{h,1})\\rightarrow S_\\partial(P\\times W_{h+g,1})\\]\nis an equivalence. In the setting of \\cref{thm:2-type-invariance-detailed}, this map is induced by the inclusion $(\\mathrm{id}_P\\times \\mathrm{inc})\\colon P\\times W_{h,1}\\hookrightarrow P\\times W_{h+g,1}$, so it is an equivalence by part \\ref{enum:2-type-iii} of the theorem because the latter inclusion is an equivalence on tangential $2$-types by \\cref{rem:2-connected-maps} since we assumed $2n\\ge4$.\n\nTo show \\ref{enum:group-completion-ii}, we first recall from \\cref{sec:disc-structure-spaces} that $\\pi_0\\, S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(P\\times D^{2n})$ is the set of equivalence classes of pairs $(W,\\phi)$ of a compact manifold $W$ whose boundary is identified with $P\\times S^{2n-1}$, together with an equivalence $\\phi\\colon E_M\\rightarrow E_{P\\times D^{2n}}$ in $\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{P\\times S^{2n-1}}$, and two such pairs are equivalent if there exists a diffeomorphism between the manifolds that makes the evident triangle in $\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{P\\times S^{2n-1}}$ homotopy commute. Forgetting $\\phi$ induces an exact sequence of pointed sets\n\\begin{equation}\\label{eqn:two-homomorphisms}\n\t\t\\pi_0\\,\\mathrm{Aut}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)^{\\simeq}_{E_{P \\times S^{2n-1} \\times I}}}(E_{P \\times D^{2n}}) \\longrightarrow \\pi_0\\,S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(P \\times D^{2n}) \\longrightarrow M_\\partial(P \\times D^{2n}) \\longrightarrow 0,\n\\end{equation}\nwhere $M_\\partial(P \\times D^{2n})$ is the pointed set of compact $d$-manifolds $W$ with boundary identified with $P \\times S^{2n-1}$ such that there exists an unspecified equivalence $E_W \\simeq E_{P \\times D^{2n}}$ in $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P \\times D^{2n}}}$, up to diffeomorphism relative to the boundary, and based at $P \\times D^{2n}$. The monoid structure on $\\pi_0\\,S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(P \\times D^{2n})$ given by the ``pair of pants''-product induces a compatible monoid structure on $M_\\partial(P \\times D^{2n})$, concretely given by\n\\begin{equation}\\label{equ:pair-of-pants}\n\tW \\Ydown W' \\coloneqq (W \\sqcup W') \\cup_{P \\times S^{2n-1} \\sqcup P \\times S^{2n-1}} P \\times W_{0,2+1}.\n\\end{equation}\nA priori, the leftmost pointed set in \\eqref{eqn:two-homomorphisms} carries \\emph{two} monoid structures---one induced by the ``pair of pants''-product $(-)\\Ydown(-)$ and one by composition---but these agree by the Eckmann--Hilton argument. Thus \\eqref{eqn:two-homomorphisms} is an exact sequence of monoids whose leftmost term is a group. Monoid-extensions of groups are groups, so it suffices to show that $M_\\partial(P \\times D^{2n})$ is a group. We do so by showing that every element has a right- and a left-inverse; since the two constructions are essentially identical, we will only explain the right-inverse. \n\t\nFor this it is convenient to use the notion of \\emph{relative bordism} $V \\colon N_0 \\leadsto N_1$ between two compact manifolds $N_0$ and $N_1$ with identified boundary $\\partial N_0\\cong \\partial N_1$, by which we mean a compact manifold $V$ with a division of its boundary into three codimension zero submanifolds $\\partial V = N_0 \\cup (\\partial N_0 \\times I) \\cup N_1$ that intersect at corners. Up to creating some corners, we can regard an element $W\\in M_\\partial(P\\times D^{2n})$ as a relative bordism of the form $P \\times D^{2n-1} \\leadsto P \\times D^{2n-1}$ by dividing the identified boundary $\\partial W\\cong P\\times S^{2n-1}$ into $(P \\times D^{2n-1}\\times\\{0\\}) \\cup (P \\times \\partial D^{2n-1}\\times [0,1]) \\cup (P \\times D^{2n-1}\\times\\{1\\})$. In these terms, the monoid structure is given by composition of relative bordisms. By definition of $M_\\partial(P\\times D^{2n})$, the manifold $W$ admits an equivalence $\\phi \\colon E_W \\to E_{P \\times D^{2n}}$ in $\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{P\\times S^{2n-1}\\times I}}$. In general, for manifold $N$ viewed as a nullbordism $N\\colon \\varnothing\\leadsto \\partial N$, we can consider the composition  \\[E_{\\partial N\\times I}\\simeq E_{\\varnothing}\\otimes E_{\\partial N\\times I}\\xra{E_{\\iota}\\otimes \\mathrm{id}} E_{N}\\otimes E_{\\partial N\\times I}\\xra{\\text{act}} E_N\\]\nin $\\ensuremath{\\mathrm{PSh}}(\\ensuremath{\\icat{D}\\mathrm{isc}}_d)$ using the Day convolution product $\\otimes$, the unique embedding $\\varnothing\\rightarrow N$ and the fact that $E_{\\varnothing}$ is the monoidal unit. Evaluating this composition at $\\ensuremath{\\mathbf{R}}^d$ and taking quotients by the $\\ensuremath{\\mathrm{Diff}}(\\ensuremath{\\mathbf{R}}^d)\\simeq\\ensuremath{\\mathrm{Emb}}(\\ensuremath{\\mathbf{R}}^d,\\ensuremath{\\mathbf{R}}^d)$-action by functoriality recovers the homotopy class of the boundary inclusion $\\partial N\\subset N$. Applying this principle to the equivalence $\\phi$ above, we obtain a homotopy equivalence $W \\simeq P \\times D^{2n}$ under $P \\times S^{2n-1}$. In terms of relative bordisms, this says that $W$ is a \\emph{strongly inertial} relative $h$-cobordism: that is, not only are the inclusions of the incoming and outgoing boundary homotopy equivalences, but the induced homotopy equivalence between them is homotopic to a diffeomorphism relative to the boundary. Since we assumed $d \\geq 6$, relative $h$-cobordisms $W \\colon W_0 = P \\times D^{2n-1} \\leadsto W_1$, up to diffeomorphism relative to the incoming boundary $P \\times D^{2n-1}$, are classified by their Whitehead torsion $\\tau(W) \\in \\mathrm{Wh}_1(\\pi_1\\,P)$, and the Whitehead torsions of strongly inertial relative $h$-cobordisms form a subgroup (cf.\\,the discussion in \\cite[Section 3]{JahrenKwasik}). Thus we may find another strongly relative $h$-cobordism $W' \\colon P \\times D^{2n-1} \\leadsto P \\times D^{2n-1}$ with a diffeomorphism $W \\cup_{P \\times D^{2n-1}} W' \\cong P \\times D^{2n}$ that respects part of the boundary identification, namely $P \\times D^{2n-1} \\{0\\}\\cup (P \\times \\partial D^{2n-1} \\times [0,1])$. By changing the identification of the outgoing boundary of $W'$ if necessary, we may assume that this diffeomorphism respects the full boundary identification. Smoothing corners, this gives a diffeomorphism $\\psi \\colon W \\Ydown W' \\to P \\times D^{2n}$ relative to $P \\times S^{2n-1}$. To show that $W'$ is a right inverse to $W$ in $M_\\partial(P\\times D^{2n})$ it thus suffices to produce an equivalence $E_{W'}\\simeq E_{P \\times D^{2n}}$ in $\\ensuremath{\\icat{M}\\mathrm{od}}_{E_{P \\times S^{2n-1} \\times I}}$. This is given by: \\vspace{-0.2cm}\n\\[\n\tE_{W'} \\simeq E_{P \\times D^{2n}} \\Ydown E_{W'} \\xrightarrow{\\phi^{-1} \\Ydown \\mathrm{id}} E_W \\Ydown E_{W'} \\simeq E_{W \\Ydown W'} \\xrightarrow{E_\\psi} E_{P \\times D^{2n}}.\\qedhere\n\\]\n\\end{proof}\n\n\\begin{cor}\\label{cor:infinite-loop-space-for-products}For $4\\le 2n\\le d$ with $d\\ge6$ and a closed $(d-2n)$-manifold $P$, the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(P\\times D^{2n})$ admits the structure of an infinite loop space.\n\\end{cor}\n\n\\begin{proof}Combining both parts of \\cref{prop:group-completion}, $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(P\\times D^{2n})$ agrees with the fibre of the group completion of the right vertical map in \\eqref{equ:sdisc-as-pullback}. As the group completion of a map of $\\ensuremath{\\catsingle{W}}$-algebras, this map can be enhanced to a map of infinite loop spaces by \\cref{thm:stability-operads} and \\cref{prop:manifold-operad-w-homstab}. Fibres of infinite loop maps carry infinite loop space structures, so the claim follows. \n\\end{proof}\n\nCombining \\cref{cor:infinite-loop-space-for-products} with the invariance under the tangential $2$-type from \\cref{thm:2-type-invariance-detailed}, we can complete the goal of this section: \n\n\\begin{proof}[Proof of \\cref{thm:oo-loop-general}] For $M$ a compact $d$-manifold with $d\\ge8$, we pick an $2n\\ge4$ such that $2n-d\\ge4$ (the choice $2n=4$ always works) and use the case $k=2$ of \\cref{lem:nice-representative-k-type} to pick a closed $(d-2n)$-manifold $P$ of the same tangential $2$-type as $M$. Both $P\\times D^{2n}$ and $M$ are $d$-dimensional and have the same tangential $2$-type, so $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(P\\times D^{2n})\\simeq S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ by \\cref{thm:2-type-invariance-detailed} \\ref{enum:2-type-i}. As $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(P\\times D^{2n})$ admits the structure of an infinite loop space by \\cref{cor:infinite-loop-space-for-products}, the first part  follows. For the claimed improvement, one can replace the role of $P$ in the argument with $P=S^{d-2n}$ for any $2n\\ge4$ with $d-2n\\ge2$, using that any two $1$-connected spin manifolds have the same tangential $2$-type (see \\cref{rem:classification-2-types}).\n\\end{proof}\n\n\\begin{rem}The construction of the infinite loop space structure on $\\ensuremath{\\catsingle{S}}_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(M)$ as presented in this section comes with several drawbacks:\n\\begin{enumerate}\n\t\\item It depends on several choices, most notably: (a) the choice of $2n\\ge4$ with $2n-d\\ge4$ and (b) the choice of a closed $(d-2n)$-manifold $P$ of the same tangential $2$-type as $M$.\n\t\\item The restrictions on the dimension are likely not optimal.\n\t\\item The space $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(P\\times D^{2n})$ ought to carry the structure of an $E_{2n}$-algebra and the infinite loop space structure we give ought to extend this $E_{2n}$-structure.\n\\end{enumerate}\n\\end{rem}\n\n\n\\section{Localisations of mapping spaces between operads} \\label{sec:ed-operads} \nThis section serves to prove general results on mapping spaces between (truncated) operads and their localisations at collections of primes. In particular, given $\\infty$-operads $\\ensuremath{\\catsingle{O}}$ and $\\ensuremath{\\catsingle{P}}$, we rely on work of G\\\"oppl \\cite{Goppl} to study the effect on homotopy groups of a map\n\\begin{equation}\\label{equ:t-localisation-mapping-spaces}\n\t\\mathrm{Map}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})_\\ensuremath{\\mathbf{Q}}\\rightarrow \\mathrm{Map}(\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}})\n\\end{equation}\nfrom the rationalisation of the mapping space between $\\ensuremath{\\catsingle{O}}$ and $\\ensuremath{\\catsingle{P}}$ to the mapping space between the respective rationalisations. In \\cref{sec:nontrivial}, we use these results to prove Theorems~\\ref{bigthm:nontrivial} and \\ref{bigcor:top-vs-auted}. \n\n\\begin{convention}\\label{conv:no-more-infty}\nUp to this point, we phrased our results and arguments in the language of $\\infty$-categories. In this and the following section, we will use several intermediate results from various sources, none of which are written in this language. To stay close to these sources, we switch language for the remainder of this paper and work in the category of simplicial sets or the category of compactly generated weak Hausdorff spaces. We denote either of these categories by $\\cat{S}$ and leave the necessary transitions based on the usual Quillen equivalence between the standard model structures on these categories to the reader. As a result of not working $\\infty$-categorically, we have to derive all mapping spaces in various categories that appear (spaces, operads, etc.) with respect to a class of weak equivalences, e.g.\\,using Dwyer--Kan's functorial simplicial localisation \\cite{DwyerKanFunction,DwyerKanSimplicial}. We indicate various derived mapping spaces by adding an $h$-subscript, so write $\\mathrm{Map}^h(-,-)$, and we will mention the class of weak equivalences with respect we derive whenever a new type of derived mapping space is considered.\n\\end{convention}\n\n\n\\subsection{Localisation of spaces and groups at a set of primes}\\label{sec:localisation}\nWe first recall some facts about $T$-localisations of spaces for a set of primes $T$. Recall that a space $Z$ is \\emph{$T$-local} if the map $(- \\circ g) \\colon \\mathrm{Map}^h_\\cat{S}(Y,Z) \\to \\mathrm{Map}^h_\\cat{S}(X,Z)$ is a weak equivalence for any map $g \\colon X \\to Y$ that is an isomorphism on $\\ensuremath{\\mathrm{H}}_*(-;\\ensuremath{\\mathbf{Z}}_T)$. Here the mapping spaces are derived with respect to  weak homotopy equivalences, and $\\ensuremath{\\mathbf{Z}}_T$ is the localisation of $\\ensuremath{\\mathbf{Z}}$ obtained by inverting all primes in $T$. Immediately from the definition, we see that the class of $T$-local spaces is closed under\n\\begin{enumerate}[(i)]\n\t\\item taking homotopy limits,\n\t\\item passing to collections of path components,\n\t\\item applying $\\mathrm{Map}^h_\\cat{S}(X,-)$ for any space $X$.\n\\end{enumerate}\nA map $f \\colon X \\to Y$ is a \\emph{$T$-localisation} if $Y$ is $T$-local and $f$ is a $\\ensuremath{\\mathbf{Z}}_T$-homology isomorphism. Any space admits a $T$-localisation and suitably modelled, this yields an $\\cat{S}$-enriched functor \n\\begin{equation}\n\t\\label{equ:localisation} \\gls*{tloc} \\colon \\cat{S} \\longrightarrow \\cat{S}\n\\end{equation}\ntogether with a natural transformation $r_T \\colon \\mathrm{id} \\to (-)_T$ which enjoys the following properties (see e.g.\\ \\cite[1.A.3, 1.A.8, 1.B.2, 1.B.7, 1.C.9,  1.C.13, 1.E.4]{Farjoun}):\n\\begin{enumerate}[(a)]\n\t\\item \\label{enum:T-localisation} the map $r_T \\colon X \\to X_T$ is a $T$-localisation, so a weak equivalence if $X$ is $T$-local,\n\t\\item \\label{enum:weak-equ-T-localisation}$(-)_T$ preserves weak equivalences,\n\t\\item \\label{enum:product-T-localisation} the canonical map $(X\\times Y)_T\\rightarrow X_T\\times Y_T$ is a weak equivalence,\n\t\\item \\label{enum:precomp-T-localisation}the map $(-)\\circ r_T\\colon \\mathrm{Map}^h_\\cat{S}(X_T,Y)\\rightarrow \\mathrm{Map}^h_\\cat{S}(X,Y)$\nis a weak equivalence if $Y$ is $T$-local.\n\\end{enumerate}\nIf $T$ is the set of all primes, $T$-localisation is rationalisation which we denote as $\\gls*{ratloc}$.\n\n\\subsubsection{Localisation of groups}\\label{sec:localisation-groups}\nRecall that a group $G$ is \\emph{$T$-local} if the map $(- \\circ g) \\colon \\mathrm{Hom}(H,G) \\to \\mathrm{Hom}(K,G)$ is an isomorphism for all $g \\colon K \\to H$ such that $\\ensuremath{\\mathrm{H}}_1(g;\\ensuremath{\\mathbf{Z}}_T)$ is an isomorphism and $\\ensuremath{\\mathrm{H}}_2(g;\\ensuremath{\\mathbf{Z}}_T)$ is surjective. The homotopy groups of a $T$-local space at any basepoint are $T$-local groups \\cite[Theorem 5.5]{Bousfield}. A morphism of groups $f\\colon H\\rightarrow G$ is a $T$-localisation if $G$ is $T$-local and $f$ has the property on $\\ensuremath{\\mathrm{H}}_*(-;\\ensuremath{\\mathbf{Z}}_T)$ for $*=1,2$ just described. One way to construct $T$-localisations of groups is as follows: the functor \\eqref{equ:localisation} has an analogue $(-)_T \\colon \\cat{S}_* \\rightarrow \\cat{S}_*$ in the pointed setting, which agrees with \\eqref{equ:localisation} on connected spaces \\cite[A.7]{Farjoun}. Defining $G_T\\coloneqq \\pi_1((BG)_T)$, we obtain a functor $(-)_T \\colon \\cat{Grp} \\rightarrow \\cat{Grp}$ on the category of groups with a natural transformation $\\mathrm{id}\\rightarrow (-)_T$ which is a $T$-localisation \\cite[Lemma 7.3]{Bousfield}. Note that we have $(G)^{\\mathrm{ab}}\\otimes\\ensuremath{\\mathbf{Z}}_T\\cong (G_T)^{\\mathrm{ab}}\\otimes\\ensuremath{\\mathbf{Z}}_T$ by construction and the Hurewicz theorem. On nilpotent groups, $(-)_T$ agrees with the usual $T$-localisation of nilpotent groups in the algebraic sense.\n\n\\subsubsection{Localisation of nilpotent spaces}\nRecall that a space $X$ is \\emph{nilpotent} if it is connected, has nilpotent fundamental group, and its $\\pi_1(X)$-action on $\\pi_i(X)$ for $i\\ge2$ is nilpotent. $T$-localisation preserves nilpotent spaces and can be characterised as follows  (see e.g.\\,\\cite[6.1.2]{MayPonto}):\n\n\\begin{lem}Let $f \\colon X \\to Y$ be a map from a nilpotent space $X$ to a $T$-local space $Y$. Then the following are equivalent:\n\t\\begin{enumerate}\n\t\t\\item $f \\colon X \\to Y$ is a $T$-localisation of spaces,\n\t\t\\item $f_* \\colon \\widetilde{\\ensuremath{\\mathrm{H}}}_k(X;\\ensuremath{\\mathbf{Z}}) \\to \\widetilde{\\ensuremath{\\mathrm{H}}}_k(Y;\\ensuremath{\\mathbf{Z}})$ is a $T$-localisation of abelian groups for all $k\\ge1$,\n\t\t\\item $f_* \\colon \\pi_k(X) \\to \\pi_k(Y)$ is a $T$-localisation of abelian and nilpotent groups for all $k\\ge1$.\n\t\\end{enumerate} \n\\end{lem}\n\nLocalisations of nilpotent spaces behave well with respect to many constructions, such as:\n\n\\begin{lem}\\label{lem:lem-hopb}Let $f \\colon X \\to A$ and $g \\colon Y \\to A$ be based maps between spaces with nilpotent basepoint component. Then\n\t\\begin{enumerate}\n\t\t\\item \\label{enum:loc-hopb-nilpotent} the basepoint component $(X\\times^h_AY)_0\\subseteq X\\times^h_AY$ of the homotopy pullback is nilpotent,\n\t\t\\item \\label{enum:loc-hopb-loc} the natural map $(X\\times^h_AY)_0 \\to (X_T\\times^h_{A_T}Y_T)_0$ is a $T$-localisation of nilpotent spaces, and\n\t\t\\item \\label{enum:loc-hopb-fg} if $X_0$, $Y_0$, and $A_0$ have finitely generated homotopy groups then so does $(X\\times^h_AY)_0$.\n\t\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}Since $(X_0\\times^h_{A_0}Y_0)_0=(X\\times^h_AY)_0$ and similarly for the localised version, we may assume that $X$, $Y$, and $A$ are connected. In this case, \\ref{enum:loc-hopb-nilpotent} and \\ref{enum:loc-hopb-loc} are \\cite[6.2.5]{MayPonto}. For \\ref{enum:loc-hopb-fg}, we use the long exact sequence for the homotopy groups of a homotopy pullback which exhibits $\\pi_i(X\\times^h_AY)$ for $i\\ge1$ as a central extension of subquotients of finitely generated nilpotent groups. As the latter are closed under taking subgroups, quotients, and extensions, the statement follows.\n\\end{proof}\n\nThe next lemma involves equivariant mapping spaces $\\mathrm{Map}_{G}(-,-) \\coloneqq \\mathrm{Map}_{\\cat{S}^G}(-,-)$ between $G$-spaces for finite groups $G$ which we derive with respect to the $G$-equivariant maps whose underlying maps of spaces are weak homotopy equivalences.\n\n\\begin{lem}\\label{lem:equivariant-mapping-spaces-nilpotent}Let $X$ and $Y$ be $G$-spaces for $G$ a finite group. If\n\\begin{itemize}\n\\item $X_{hG}$ is weakly equivalent to a finite CW complex and \n\\item $Y$ has nilpotent path components,\n\\end{itemize}\nthen for any $f\\in\\mathrm{Map}^h_G(X,Y)$ the following holds:\n\\begin{enumerate}\n\\item the path component $\\mathrm{Map}^h_G(X,Y)_f\\subseteq \\mathrm{Map}^h_G(X,Y) $ is nilpotent,\n\\item the postcomposition map $(r_T \\circ (-))\\colon \\mathrm{Map}^h_G(X,Y)_f\\rightarrow\\mathrm{Map}^h_G(X,Y_T)_{(r_T\\circ f)}$ is a $T$-localisation,\n\\item\\label{enum:Y-finitely-gen-homotopy} if $Y$ has finitely generated homotopy groups at all basepoints, then so does $\\mathrm{Map}^h_G(X,Y)_f$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nBy the assumption on $X_{hG}$, we may assume that $X$ is a finite $G$-CW complex consisting of free $G$-cells. This allows us to argue by induction on the number of cells: if $X$ is obtained from $X'$ by attaching a single free $G$-cell, there are commutative squares\n\\[\\begin{tikzcd} \n\tS^{d-1} \\times G \\rar \\dar & X' \\dar \\\\\n\tD^{d-1} \\times G \\rar & X\\end{tikzcd}\\qquad\\text{and}\\qquad\\begin{tikzcd} \\mathrm{Map} ^h_G(X,Y) \\rar \\dar & \\mathrm{Map} ^h(D^d,Y) \\dar \\\\\n\t\\mathrm{Map} ^h_G(X',Y) \\rar & \\mathrm{Map} ^h(S^{d-1},Y).\n\\end{tikzcd}\\]\nThe left square is a homotopy pushout of $G$-spaces and the right square is obtained from it by applying $\\mathrm{Map} ^h_G(-,Y)$, so it is a homotopy pullback. By an induction over a principal Postnikov tower of the path components of $Y$, one sees that the conclusions hold for all components of the right-hand terms of the right-hand diagram. By induction, we may assume they hold for the bottom-left corner in the right-hand diagram, so by an application of \\cref{lem:lem-hopb} and using that subgroups of (finitely generated) nilpotent groups are (finitely generated) nilpotent, they also hold for all components of the top-left corner in the right-hand diagram.\n\\end{proof}\n\nRecall that a \\emph{$\\ul{k}$-cubical diagram} is a functor on the poset of subsets of $\\ul{k}\\coloneqq \\{1,\\ldots,k\\}$.\n\n\\begin{lem}\\label{lem:nilpotent-cube}\nLet $X$ be a $\\ul{k}$-cubical diagram of spaces with nilpotent path components.\n\\begin{enumerate}\n\t\\item $\\ensuremath{\\mathrm{holim}}_{\\varnothing\\neq I\\subseteq \\ul{r}}X(I)$ has nilpotent components and the map $\\ensuremath{\\mathrm{holim}}_{\\varnothing\\neq I\\subseteq \\ul{k}}X(I)\\rightarrow \\ensuremath{\\mathrm{holim}}_{\\varnothing\\neq I\\subseteq \\ul{k}}(X(I)_T)$ induced by the $T$-localisations of the $X(I)$'s, is a $T$-localisation when restricted to any component of the source and the corresponding component of the target.\n\t\\item If $X(I)$ has finitely generated homotopy groups at all basepoints for all $\\varnothing \\neq I\\subseteq \\ul{k}$, then $\\ensuremath{\\mathrm{holim}}_{\\varnothing\\neq I\\subseteq \\ul{k}}X(I)$ has finitely generated homotopy groups at all basepoints.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nWe prove the claim by induction on $k$. For $k=1$  the claim is vacuous as  $\\ensuremath{\\mathrm{holim}}_{\\varnothing\\neq I\\subseteq \\ul{1}}X(I)\\simeq X(\\underline{1})$. For larger $k$, we use that the homotopy limit fits into a homotopy cartesian square\n\\[\\begin{tikzcd}\n\t\\underset{\\varnothing\\neq I\\subseteq \\ul{k}}\\ensuremath{\\mathrm{holim}}\\, X(I)\\rar\\dar&X(\\ul{k})\\dar\\\\\n\t\\underset{\\varnothing\\neq I\\subseteq \\ul{k-1}}\\ensuremath{\\mathrm{holim}}\\,X(I)\\rar& \\underset{\\varnothing\\neq I\\subseteq \\ul{k-1}}\\ensuremath{\\mathrm{holim}}\\,X(I\\cup\\{k\\}).\n\\end{tikzcd}\\]\n By induction, the conclusion of the statement holds for two diagrams defining the bottom row, and by assumption also for $X(\\ul{k})$, so \\cref{lem:lem-hopb} gives the induction step.\n\\end{proof}\n\n\\subsection{Operads and dendroidal spaces}\\label{section:operads}In this and the following sections, \\emph{operads} $\\cat{O}, \\cat{P},\\ldots$ are understood as single-coloured operads in $\\ensuremath{\\mathbf{S}}$ in the classical sense. Declaring a weak equivalence to be a levelwise weak equivalence gives rise to derived mapping spaces $\\mathrm{Map}_{\\cat{Opd}}^h(\\cat{O},\\cat{P})$ between such operads. Under mild assumptions on $\\cat{O}$ and $\\cat{P}$, there are two equivalent point of views on these mapping spaces that we will make us of, related by natural maps\n\\begin{equation}\\label{equ:different-models-mapping-spaces}\n\t\\mathrm{Map}_{\\cat{Opd}}^h(\\cat{O},\\cat{P})\\xlra{\\circled{1}}\\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{De})}(N_d \\cat{O},N_d \\cat{P})\\xlra{\\circled{2}}\\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{rDe})}(N_d \\cat{O},N_d \\cat{P})\n\\end{equation}\nwhich we explain in the following two subsections. Part of our discussion in this and the following subsection is similar to that in \\cite[Section 3.4]{WeissDalian}.\n\n\\subsubsection{Dendroidal spaces and the map $\\circled{1}$}\nThe two alternative points of view stem from Moerdijk--Weiss' \\emph{dendroidal spaces}. Briefly (see \\cite{MoerdijkWeiss} for details), the category of \\emph{dendroidal spaces} is the category of presheaves $\\ensuremath{\\catsingle{O}} \\colon {\\cat{De}}^{\\mathrm{op}} \\to \\cat{S}$ on a certain category $\\gls*{dend}$ of trees. The category ${\\cat{De}}$ has as objects finite rooted trees with specified subsets of leaves. More formally an object $(t,\\le,\\ell(t))$ in $\\cat{De}$ is a finite partially ordered set $(t,\\le)$ of \\emph{edges} together with a specified subset $\\ell(t)\\subset t$ of maximal elements (the \\emph{leaves}) such that (a) $\\{v\\in t\\,|\\,v\\le w\\}$ is totally ordered for all $w\\in T$ and (b) there is an element $v\\in T$ (the \\emph{root}) such that $v\\le w$ for all $w\\in T$. The subset $\\nu(t)\\coloneqq t\\backslash \\ell(T)\\subset t$ is the set of \\emph{vertices} of the tree. The \\emph{incoming edges} $\\mathrm{in}(v)\\subset t$ of a vertex $v$ is the set of minimal elements in $\\{w\\in t\\,|\\,w>v\\}$. Morphisms in $\\cat{De}$ can be graphically viewed as compositions of edge expansions, leaf additions, and deleting of bivalent vertices; see Section 3 loc.cit.\\ for details. There is a functor $N_d(-)$ from operads to dendroidal spaces called the \\emph{dendroidal nerve} \\cite[Example 4.2]{MoerdijkWeiss}, given by\n\\[\n\t\\textstyle{\\gls*{dendn} \\cat{O}(t)\\coloneqq  \\bigsqcap_{v\\in \\nu(t)} \\cat{O}(|\\mathrm{in}(v)|)}.\n\\]\nDeclaring weak equivalences between dendroidal spaces to be levelwise weak equivalences gives rise to derived mapping spaces $\\gls*{mapdend}$ of dendroidal spaces and as $N_d(-)$ clearly preserves weak equivalence, we obtain the map $\\circled{1}$. \n\n\\subsubsection{Dendroidal Segal spaces}\nThere is a convenient class of dendroidal spaces that includes those coming from operads but is homotopically more flexible. To define it, we consider the \\emph{$k$-corolla} which is the unique (up to isomorphism) tree in ${\\cat{De}}$ with one vertex and $k$ leaves, denoted by $t_k$. The unique (up to isomorphism) tree in ${\\cat{De}}$ with no vertices is denoted $\\eta$. For each vertex $v$ in a tree $t$, there is a morphism $t_k\\rightarrow t$ (unique up to automorphism of $t_k$) that takes the root to $v$ and the leaves to $\\mathrm{in}(v)$. Given a dendroidal space $\\ensuremath{\\catsingle{O}}$ and a tree $t$, these morphisms assemble to a map\n\\begin{equation}\\label{equ:dendroidal-segal-maps}\n\t\\textstyle{\\ensuremath{\\catsingle{O}}(t)\\longrightarrow\\bigsqcap_{v\\in \\nu(t)}\\ensuremath{\\catsingle{O}}(t_{|\\mathrm{in}(v)|})}.\n\\end{equation}\n\n\\begin{dfn}\nA \\emph{dendroidal Segal space} $\\ensuremath{\\catsingle{O}}$ is a dendroidal space such that \\eqref{equ:dendroidal-segal-maps} is a weak equivalences for all trees $t\\in \\cat{De}$. This says in particular that $\\ensuremath{\\catsingle{O}}(\\eta)$ is weakly contractible.\n\\end{dfn}\n\n\\begin{rem}\nThis definition agrees with \\cite[Definition 4.1]{BGR}. Under the assumption that $\\ensuremath{\\catsingle{O}}(\\eta)$ is weakly contractible, it also agrees with \\cite[Definition 1.2.9]{Goppl}. The original definition of a dendroidal Segal space \\cite[Definition 5.4]{CisinskiMoerdijkSegal} assumes additional fibrancy conditions.\n\\end{rem}\n\n\\subsubsection{Reduced dendroidal spaces and the map $\\circled{2}$}\nThe full subcategory $\\gls*{rdend} \\subset {\\cat{De}}$ of \\emph{reduced trees}, i.e.\\ trees $t$ with $\\ell(v)=\\varnothing$, is often easier to work with. Presheaves $\\ensuremath{\\catsingle{O}} \\colon {\\cat{rDe}}^{\\mathrm{op}} \\to \\cat{S}$ are called \\emph{reduced dendroidal spaces}. Morphisms between those are still natural transformations and weak equivalences are levelwise; we denote the resulting derived mapping spaces by $\\gls*{maprdend}$. Restriction along ${\\cat{rDe}}\\subset {\\cat{De}}$ induces a natural map \n\\[\\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{De})}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})\\longrightarrow \\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{rDe})}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})\\]\nof which $\\circled{2}$ is a special case.\n\n\\subsubsection{Comparisons}\nBy work of Boavida de Brito--Weiss and a variant of a result of Cicinski--Moerdijk we learned from work of Boavida de Brito--Horel--Robertson, both maps in \\eqref{equ:different-models-mapping-spaces} are weak equivalences as long as the operads $\\cat{O}$ and $\\cat{P}$ are \\emph{$1$-reduced}, i.e.\\,if their spaces of $0$- and $1$-ary operations are weakly contractible.\n\n\\begin{prop}\\label{prop:comparison-of-models}The map $\\circled{1}$ is a weak equivalence for all operads $\\cat{O}$ and $\\cat{P}$. If $\\cat{O}$ and $\\cat{P}$ are $1$-reduced, then also $\\circled{2}$ is a weak equivalence.\n\\end{prop}\n\n\\begin{proof}\nFor the map $\\circled{1}$, this is \\cite[Proposition 4.3]{BGR}. For $\\circled{2}$, the proof follows a straight-forward adaptation of the proof of \\cite[Lemma 7.12]{BdBWConf}. \n\\end{proof}\n\n\n\\subsection{A tower of derived mapping spaces}\\label{sec:goppl-tower}\nThe category ${\\cat{rDe}}$ admits a filtration\n\\[\n\t{\\cat{rDe}}_{\\leq 0} \\subset {\\cat{rDe}}_{\\leq 1} \\subset \\cdots \\subset {\\cat{rDe}},\n\\]\nby the full subcategories $\\gls*{rdendk}$ on those trees whose vertices $v$ have at most $k$ incoming edges. Denoting the restriction of a reduced dendroidal space $\\ensuremath{\\catsingle{O}}$ along ${\\cat{rDe}}_{\\leq k}\\subset {\\cat{rDe}}$ by the same symbol, we obtain a natural tower of derived mapping spaces\n\\begin{equation}\\label{equ:tower}\n\t\\begin{tikzcd}[row sep=0.6cm,column sep=2cm]\n\t\t&\\ldots \\dar\\\\\n\t\t&\\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{rDe}_{\\le1})}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})\\dar\\\\\n\t\t\\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{rDe})}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})\\arrow[ur]\\arrow[uur]\\rar&\\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{rDe}_{\\le0})}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}),\n\t\\end{tikzcd}\n\\end{equation}\nall derived with respect to the levelwise weak equivalences. For simplicity we write\n\\begin{equation}\\label{eqn:abbrevations-op-maps}\n\t\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})\\coloneqq \\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{rDe})}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})\\quad\\text{and}\\quad \\mathrm{Map}^h_{\\leq k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})\\coloneqq \\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{rDe}_{\\le k})}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}).\n\\end{equation}\nThis tower was studied by G\u00f6ppl \\cite{Goppl}. In Lemma 2.1.1 loc.cit.\\,he notes that it \\emph{converges}, that is, we have a weak equivalence\n\\begin{equation}\\label{equ:goppl-convergence}\n\t\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})\\xlra{\\simeq} \\underset{k}\\ensuremath{\\mathrm{holim}}\\,\\mathrm{Map}^h_{\\leq k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}).\n\\end{equation} \nTo identify its \\emph{layers}, i.e.\\ the homotopy fibres of the vertical maps, G\u00f6ppl considers the $k$th \\emph{matching} and \\emph{latching} object of a reduced dendroidal space $\\ensuremath{\\catsingle{O}}$\n\\[\n\t\\gls*{latchk} \\coloneqq \\underset{(\\overline{t}_k\\rightarrow t) \\in ({\\cat{rDe}}_{\\leq k-1})_{\\overline{t}_k\/}}\\mathrm{hocolim} \\, \\ensuremath{\\catsingle{O}}(t),\\quad \\text{and} \\quad\n\t\\gls*{matchk} \\coloneqq \\underset{{( t\\rightarrow \\overline{t}_k) \\in ({\\cat{rDe}}_{\\leq k-1})_{\/\\overline{t}_k}}}\\ensuremath{\\mathrm{holim}}\\,  \\ensuremath{\\catsingle{P}}(t).\n\\]\nHere $\\overline{t}_k\\in \\cat{rDe}$ is the \\emph{reduced $k$-corolla}, the unique (up to isomorphism) reduced tree with $k+1$ vertices of which one has $k$ incoming edges and the others have none. Permuting incoming edges defines an action of the symmetric group $\\Sigma_k$ on $\\overline{t}_k$ in $\\cat{rDe}$ which induces a natural $\\Sigma_k$-action on $\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}})$ and $\\mathrm{Latch}_k(\\ensuremath{\\catsingle{P}})$. These are related by $\\Sigma_k$-equivariant maps \n\\begin{equation}\\label{equation:latch-match}\n\t\\mathrm{Latch}_k(\\ensuremath{\\catsingle{O}})\\longrightarrow \\ensuremath{\\catsingle{O}}(\\overline{t}_k)\\longrightarrow \\mathrm{Match}_k(\\ensuremath{\\catsingle{O}}).\n\\end{equation} \nFor $\\ensuremath{\\catsingle{O}}$ and $\\ensuremath{\\catsingle{P}}$ dendroidal Segal spaces, G\u00f6ppl uses these maps to identify the vertical homotopy fibres in the above tower in terms of the matching and latching objects and spaces of derived maps between $\\Sigma_k$-spaces, see Theorem 2.1.14 and Remark 2.1.22 loc.cit., under the (mild) assumption that $\\ensuremath{\\catsingle{O}}$ and $\\ensuremath{\\catsingle{P}}$ are \\emph{$1$-reduced}, i.e.\\ the value at the $0$- and $1$-corolla are weakly contractible (recall that we assume the value at the tree with no vertex is weakly contractible). \n\n\\begin{thm}[G\u00f6ppl]\\label{thm:goppl}\nFor $k\\ge1$ and $1$-reduced dendroidal Segal spaces $\\ensuremath{\\catsingle{O}}$ and $\\ensuremath{\\catsingle{P}}$, there is a natural homotopy cartesian square whose left and top arrow is induced by restriction\n\\[\\begin{tikzcd} \n\t\\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}) \\dar \\rar &  \\mathrm{Map}^h_{\\Sigma_k}(\\ensuremath{\\catsingle{O}}(\\overline{t}_k),\\ensuremath{\\catsingle{P}}(\\overline{t}_k)) \\dar \\\\\n\t\\mathrm{Map}^h_{\\le k-1}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}) \\rar & P_k(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}).\n\\end{tikzcd}\\]\nThe corner $P_k(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})$ fits into a natural homotopy cartesian square\n\\[\\begin{tikzcd} \n\tP_k(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}) \\rar \\dar & \\mathrm{Map}^h_{\\Sigma_k}(\\ensuremath{\\catsingle{O}}(\\overline{t}_k),\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}})) \\dar \\\\\n\t\\mathrm{Map}^h_{\\Sigma_k}(\\mathrm{Latch}_k(\\ensuremath{\\catsingle{O}}),\\ensuremath{\\catsingle{P}}(\\overline{t}_k)) \\rar & \\mathrm{Map}^h_{\\Sigma_k}(\\mathrm{Latch}_k(\\ensuremath{\\catsingle{O}}),\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}}))\n\\end{tikzcd}\\]\n\twhose bottom and right maps are induced by \\eqref{equation:latch-match}.\n\\end{thm}\n\n\n\\subsection{Localisations of dendroidal spaces}\\label{sec:localisation-dendroidal-spaces}\n\\label{section:localisation-dendroidal-spaces}\nGiven a dendroidal space $\\ensuremath{\\catsingle{O}}$, its \\emph{$T$-localisation} $\\ensuremath{\\catsingle{O}}_T$ for a set of primes $T$ is the dendroidal  space given as the composition of $\\ensuremath{\\catsingle{O}}\\colon {\\cat{De}}^{\\mathrm{op}} \\rightarrow \\cat{S}$ with the localisation functor $(-)_T\\colon \\cat{S}\\rightarrow\\cat{S}$. The natural transformation $\\mathrm{id}_{\\cat{S}}\\rightarrow (-)_T$ induces a map $r_T\\colon \\ensuremath{\\catsingle{O}}\\rightarrow \\ensuremath{\\catsingle{O}}_T$ of dendroidal Segal spaces. It follows from properties \\ref{enum:weak-equ-T-localisation} and \\ref{enum:product-T-localisation} from \\cref{sec:localisation} that if $\\ensuremath{\\catsingle{O}}$ is a dendroidal Segal space, then so is $\\ensuremath{\\catsingle{O}}_T$, and similarly for the $1$-reduced variant. \n\n\\begin{lem}\\label{lem:loc-dendr-map} For dendroidal spaces $\\ensuremath{\\catsingle{O}}$ and $\\ensuremath{\\catsingle{P}}$ such that $\\ensuremath{\\catsingle{P}}$ is levelwise $T$-local,  $\\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{De})}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})$ is $T$-local and the natural zig-zag \n\\[\n\t\\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{De})}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}) \\xrightarrow{(-)_T} \\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{De})}(\\ensuremath{\\catsingle{O}}_T,\\ensuremath{\\catsingle{P}}_T) \\xleftarrow{(-)\\circ r_T}\\mathrm{Map}^h_{\\ensuremath{\\mathrm{PSh}}(\\cat{De})}(\\ensuremath{\\catsingle{O}}_T,\\ensuremath{\\catsingle{P}}).\n\\]\nconsists of weak equivalences. The same holds when replacing $\\cat{De}$ by $\\cat{rDe}$ or $\\cat{rDe}_{\\le k}$. \n\\end{lem}\n\n\\begin{proof}The derived mapping spaces appearing in the statement are formed in a category of space-valued presheaves with levelwise weak equivalences, so they can be computed as homotopy limits of a diagram of levelwise mapping spaces. We saw in \\cref{sec:localisation} that $T$-local spaces are closed under taking homotopy limits and applying $\\mathrm{Map}^h_\\cat{S}(X,-)$ for any space $X$, so this implies the first part of the claim. Moreover, this argument reduces the second part to proving that the zigzag of derived mapping spaces in the category of spaces\n\\[\n\t\\mathrm{Map}^h_\\cat{S}(\\ensuremath{\\catsingle{O}}(t),\\ensuremath{\\catsingle{P}}(t)) \\xrightarrow{(-)_T} \\mathrm{Map}^h_\\cat{S}(\\ensuremath{\\catsingle{O}}(t)_T,\\ensuremath{\\catsingle{P}}(t)_T) \\xleftarrow{(-)\\circ r_T} \\mathrm{Map}^h_\\cat{S}(\\ensuremath{\\catsingle{O}}(t)_T,\\ensuremath{\\catsingle{P}}(t))\n\\]\nconsists of weak equivalences for all trees $t\\in\\cat{De}$. For the second map, this follows follows the fact that $r_T\\colon \\ensuremath{\\catsingle{P}}(t)\\rightarrow \\ensuremath{\\catsingle{P}}(t)_T$ is a weak equivalence by property \\ref{enum:T-localisation} of $T$-localisation. For the first map, we note that the composition\n\\[\n\t\\mathrm{Map}^h_\\cat{S}(\\ensuremath{\\catsingle{O}}(t),\\ensuremath{\\catsingle{P}}(t))\\xra{(-)_T} \\mathrm{Map}^h_\\cat{S}(\\ensuremath{\\catsingle{O}}(t)_T,\\ensuremath{\\catsingle{P}}(t)_T)\\xra{(-)\\circ r_T} \\mathrm{Map}^h_\\cat{S}(\\ensuremath{\\catsingle{O}}(t),\\ensuremath{\\catsingle{P}}(t)_T)\n\\]\nagrees with postcomposition with $r_T\\colon \\ensuremath{\\catsingle{P}}(t)\\rightarrow \\ensuremath{\\catsingle{P}}(t)_T$, so is a weak equivalence. The second map is an weak equivalence by property \\ref{enum:precomp-T-localisation}, so the first map is one too.\n\\end{proof}\n\n\n\\subsubsection{Localisations of derived mapping spaces}\nRecalling the abbreviations of \\eqref{eqn:abbrevations-op-maps}, denoting the path component of a derived map $f\\in \\mathrm{Map}^h_{\\leq k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})$ by \n\\[\n\t\\mathrm{Map}^h_{\\leq k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})_f\\subseteq \\mathrm{Map}^h_{\\leq k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}),\n\\]\nand abbreviating $\\ensuremath{\\catsingle{O}}(\\overline{t}_k)$ to $\\ensuremath{\\catsingle{O}}(k)$, we can now state the following result:\n\n\\begin{thm}\\label{thm:truncated-operad-maps} Let $\\ensuremath{\\catsingle{P}}$ and $\\ensuremath{\\catsingle{O}}$ be $1$-reduced dendroidal Segal spaces such that for all $k\\ge0$\n\\begin{itemize}\n\t\\item $\\ensuremath{\\catsingle{P}}(k)$ has nilpotent path components and\n\t\\item $\\ensuremath{\\catsingle{O}}(k)_{h\\Sigma_k}$ and $\\mathrm{Latch}_k(\\ensuremath{\\catsingle{O}})_{h\\Sigma_k}$ are weakly equivalent to finite CW complexes,\n\\end{itemize}\nthen the following holds for all $k\\ge0$ and any map $f\\in \\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})$:\n\\begin{enumerate}[(i)]\t\t\n\t\\item \\label{enum:truncated-nilp} the path component $\\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})_f$ is nilpotent,\n\t\\item \\label{enum:truncated-loc} for a set of primes $T$, the natural map induced by $T$-localisation $r_T\\colon \\ensuremath{\\catsingle{P}}\\rightarrow \\ensuremath{\\catsingle{P}}_T$\n\t\\[\n\t\t\\mathrm{Map}^h_{\\leq k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})_f \\longrightarrow \\mathrm{Map}^h_{\\leq k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}_{T})_{r_T\\circ f}\n\t\\]\n\tis a $T$-localisation of nilpotent spaces, and\n\t\\item \\label{enum:truncated-fg} if the spaces $\\ensuremath{\\catsingle{P}}(k')$ have finitely generated homotopy groups at all basepoints for all $k'\\ge0$, then so does $\\mathrm{Map}^h_{\\leq k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})_f$.\n\\end{enumerate}\n\\end{thm}\n\nThe first part of this result (and the strategy of proof) is similar to \\cite[Proposition 5.2.4]{WeissDalian}. We start the proof with an auxiliary lemma:\n\n\\begin{lem}\\label{lem:match-nilpotent}Let $T$ be a set of primes and $\\ensuremath{\\catsingle{P}}$ a reduced dendroidal Segal space such that $\\ensuremath{\\catsingle{P}}(k)$ has nilpotent path components for all $k\\ge0$. The following holds for all $k\\ge0$:\n\\begin{enumerate}\n\t\\item $\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}})$ has nilpotent path components,\n\t\\item the natural map $\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}})\\rightarrow \\mathrm{Match}_k(\\ensuremath{\\catsingle{P}}_T)$ is a $T$-localisation when restricted to a path component of the source and the corresponding path component of the target,\n\t\\item if the spaces $\\ensuremath{\\catsingle{P}}(k')$ has finitely generated homotopy groups at all basepoints for $0 \\leq k' \\leq k$, then so does $\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}})$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nIdentifying the vertices of $\\overline{t}_k$ with no incoming edges with $\\ul{k} = \\{1,2,\\ldots,k\\}$, every subset $I\\subseteq \\ul{k}$ defines a reduced subcorolla $\\overline{t}_I\\subseteq \\overline{t}_k$. This gives rise to an $\\ul{k}$-cubical diagram $\\underline{k} \\supseteq I \\mapsto \\ensuremath{\\catsingle{P}}(\\overline{t}_{\\ul{k} \\setminus I})$.\nBy the argument above Theorem 3.4.7 in \\cite{WeissDalian}, there is a natural equivalence $\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}})\\simeq \\ensuremath{\\mathrm{holim}}_{\\varnothing\\neq I\\subseteq \\ul{k}}\\ensuremath{\\catsingle{P}}(\\overline{t}_{\\ul{k} \\setminus I})$, so the claim follows from an application of \\cref{lem:nilpotent-cube}.\n\\end{proof}\n\n\n\\begin{proof}[Proof of \\cref{thm:truncated-operad-maps}]\nWe prove the claim by induction on $k$. The initial case $k=0$ is trivial since $\\ensuremath{\\catsingle{P}}$ is assumed to be $1$-reduced, so the mapping spaces appearing in the statement are contractible. For the induction step we assume the claim for $k-1$ and prove it for $k$. To do so, we consider the homotopy cartesian squares of \\cref{thm:goppl}. A choice of $f\\in \\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})$ induces basepoints in all spaces participating in these squares; we denote these also by $f$. Now consider the maps\n\\begin{equation}\\label{equ:match-latch-localisations}\n\t\\begin{aligned}\\mathrm{Map}^h_{\\Sigma_k}(\\ensuremath{\\catsingle{O}}(k),\\ensuremath{\\catsingle{P}}(k))_f &\\longrightarrow  \\mathrm{Map}^h_{\\Sigma_k}(\\ensuremath{\\catsingle{O}}(k),\\ensuremath{\\catsingle{P}}(k)_T)_{f} \\\\\n\t\\mathrm{Map}^h_{\\Sigma_k}(\\ensuremath{\\catsingle{O}}(k),\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}}))_f &\\longrightarrow  \\mathrm{Map}^h_{\\Sigma_k}(\\ensuremath{\\catsingle{O}}(k),\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}})_T)_{f}\\\\ \n\t\\mathrm{Map}^h_{\\Sigma_k}(\\mathrm{Latch}_k(\\ensuremath{\\catsingle{O}}),\\ensuremath{\\catsingle{P}}(k))_f &\\longrightarrow  \\mathrm{Map}^h_{\\Sigma_k}(\\mathrm{Latch}_k(\\ensuremath{\\catsingle{O}}),\\ensuremath{\\catsingle{P}}(k)_T)_{ f} \\\\\n\t\\mathrm{Map}^h_{\\Sigma_k}(\\mathrm{Latch}_k(\\ensuremath{\\catsingle{O}}),\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}}))_f &\\longrightarrow  \\mathrm{Map}^h_{\\Sigma_k}(\\mathrm{Latch}_k(\\ensuremath{\\catsingle{O}}),\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}})_T)_{f} \\end{aligned}\n\\end{equation}\ninduced by postcomposition with the $T$-localisations of the codomains. Combining \\cref{lem:equivariant-mapping-spaces-nilpotent} with \\cref{lem:match-nilpotent}, all four maps are $T$-localisations of nilpotent spaces. Moreover, by the first part of \\cref{lem:match-nilpotent}, we may replace $\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}})_T$ in the codomain of the second and fourth map by $\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}}_T)$. An application of \\cref{lem:lem-hopb} to the second square in \\cref{thm:goppl} shows that the map $P_k(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})_f\\rightarrow P_k(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}_T)_{f}$ between the components induced by $f$ is a $T$-localisation of nilpotent spaces. Combining this with the induction hypothesis, another application of \\cref{lem:lem-hopb}---this time to the first square---shows that the natural map $\\smash{\\mathrm{Map}^h_{\\leq k}}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})_f \\rightarrow \\smash{\\mathrm{Map}^h_{\\leq k}}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}_{T})_{f}$ is a $T$-localisation between nilpotent spaces, so \\ref{enum:truncated-nilp} and \\ref{enum:truncated-loc} hold. \n\t\t\nWe argue similarly for \\ref{enum:truncated-fg}: if the spaces $\\ensuremath{\\catsingle{P}}(k)$ have finitely generated homotopy groups at all basepoints, then so does $\\mathrm{Match}_k(\\ensuremath{\\catsingle{P}})$ by the second part of \\cref{lem:match-nilpotent}. By the second part of \\cref{lem:equivariant-mapping-spaces-nilpotent}, we conclude that the domains of the four maps have finitely generated homotopy groups, so the same holds for $P_k(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})_f$ by an application of the final part of  \\cref{lem:lem-hopb} and thus also for $\\smash{\\mathrm{Map}^h_{\\le k}}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})_f$ by another application of that lemma and the induction hypothesis.\n\\end{proof}\n\nThis finishes the first part of this section as outlined in \\cref{sec:intr-nontriviality} after \\ref{enum:rationalisation-operads-intro}.\n\n\\subsection{Inverse limits and countability}The second part of this section begins with general results on the behaviour of homotopy groups of homotopy limits of towers of spaces.\n\n\\subsubsection{Towers of groups}\nFollowing \\cite[IX.2]{BousfieldKan}, we call a sequence of groups \n\\[\n\tG_0\\leftarrow G_1\\leftarrow G_2\\leftarrow  \\cdots\n\\] \na \\emph{tower of groups} and abbreviate it by $\\gls*{tower}$. We can assign to such a tower a limit \\emph{group} $\\lim_kG_k$ and a \\emph{pointed $\\lim^1$-set} $\\lim^1_kG_k$ \\cite[IX.2.1]{BousfieldKan}. If the tower consists of abelian groups, then $\\lim^1_k G_k$ inherits an abelian group structure. A short exact sequence of towers of groups induces a long exact sequence as follows \\cite[IX.2.3]{BousfieldKan}:\n\n\\begin{lem}\\label{lem:les} A levelwise short exact sequence of towers of groups \n\\[\n\t0\\rightarrow\\{G_k\\} \\rightarrow\\{H_k\\}\\rightarrow\\{K_k\\}\\lra0\n\\]\ninduces a natural exact sequence of groups and pointed sets\n\\[\n\t\\textstyle{0\\rightarrow \\lim_kG_k\\rightarrow\\lim_kH_k\\rightarrow\\lim_kK_k\\rightarrow \\lim^1_kG_k\\rightarrow\\lim^1_kH_k\\rightarrow\\lim^1_kK_k\\ra0.}\n\\]\n\\end{lem}\n\n\nRecall that a map $\\{f_k\\} \\colon \\{G_k\\} \\to \\{H_k\\}$ of towers of groups is called a \\emph{pro-isomorphism} if for all $s \\geq 0$ there exists a $t \\geq s$ and a homomorphism $H_{t} \\to G_s$ such that the diagram\n\\[\\begin{tikzcd} \n\tG_s \\dar[swap]{f_s} & \\lar G_{t} \\dar{f_{t}} \\\\\n\tH_s & \\lar H_{t} \\arrow{lu} \n\\end{tikzcd}\\]\ncommutes. Pro-isomorphisms have the following property \\cite[Proposition III.2.6]{BousfieldKan}:\n\n\\begin{lem}\\label{lem:pro-iso-lim} For a pro-isomorphism $\\{f_k\\} \\colon \\{G_k\\} \\to \\{H_k\\}$ the induced map $\\lim_kG_k \\to \\lim_kH_k$ is an isomorphism and the induced map $\\lim^1_kG_k \\to \\lim^1_kH_k$ is a pointed bijection.\n\\end{lem}\n\nFor a tower of groups $\\{G_k\\}$ and $r\\ge1$, the \\emph{$r$th derived tower} $(G^{(r)}_k)$ is defined by \\[G_k^{(r)}\\coloneqq \\mathrm{im}\\big(G_{k+r}\\rightarrow G_k\\big).\\]\nFor each fixed $k$, this defines a tower $\\{G^{(r)}_k\\}_{r \\in \\ensuremath{\\mathbf{N}}}$ of inclusions of subgroups. The tower $(G_k)$ is called \\emph{Mittag--Leffler} if for each $k$ there is an $m<\\infty$ so that $\\lim_{m' \\geq m} \\smash{G_k^{(m')}}\\rightarrow \\smash{G_k^{(m)}}$ is an isomorphism. Examples of Mittag--Leffler towers include towers of finite groups or finite dimensional vector spaces. Mittag--Leffler towers have the following property \\cite[Corollary IX.3.5]{BousfieldKan}:\n\n\\begin{lem}\\label{lem:ML-no-lim1} If a tower of groups $\\{G_k\\}$ is Mittag--Leffler, then $\\lim^1_kG_k=*$.\n\\end{lem}\n\nTo recognise Mittag--Leffler towers, we use the following result from \\cite[Theorem 2]{McGibbonMoller}:\n\n\\begin{lem}[McGibbon--M\u00f8ller]\\label{lem:McGibbon-Moller}\n\tFor a tower $\\{G_k\\}$ of countable groups, the following statements are equivalent:\n\t\\begin{enumerate}\n\t\t\\item $\\lim^1_k G_k$ is countable,\n\t\t\\item $\\lim^1_k G_k$ vanishes,\n\t\t\\item the tower $\\{G_k\\}$ is Mittag--Leffler.\n\t\\end{enumerate}\n\\end{lem}\n\nThe following lemma appears in \\cite[Corollary 6.1.9]{DydakSegal}, but we include a proof for the convenience of the reader. For a group $G$ we denote the constant tower with value $G$ by $\\{c\\,G\\}$.\n\n\\begin{lem}[Dydak--Segal] \\label{lem:dydak-segal} If a tower of groups $\\{G_k\\}$ is Mittag--Leffler and $\\lim_k G_k$ is countable, then the canonical map $\\{c\\lim_k G_k\\} \\to \\{G_k\\}$ is a pro-isomorphism.\n\\end{lem}\n\n\\begin{proof}Any Mittag--Leffler tower of groups $\\{G_k\\}$ is pro-isomorphic to one with surjective transition maps (consider the tower $\\{G_k'\\}$ of stable images $G_k'\\subset G_k$, i.e. $G_k'=\\mathrm{im}(G_{k+m}\\rightarrow G_k)$ for $m\\gg0$), so we may assume this is the case. This in particular ensures that the maps $\\lim_k G_k\\rightarrow G_k$ are surjective, so $G_k$ is countable for all $k\\ge0$, and it also shows that the claim is true if $\\lim_k G_k = 0$ and thus $G_k=0$ for all $k$. We use this special case to prove the following claim, which implies the general statement when applied to the map $\\{c\\lim_k G_k\\} \\to \\{G_k\\}$:\n\n\\medskip\n\n\\noindent \\textbf{Claim. } Let $\\{G_k\\}$ be a Mittag--Leffler tower of countable groups and $\\{f_k\\} \\colon \\{G_k\\} \\to \\{H_k\\}$ a levelwise surjective map of towers of groups. If $\\lim_k f_k \\colon \\lim_k G_k \\to \\lim_k H_k$ is an isomorphism, then $\\{f_k\\}$ is a pro-isomorphism.\n\n\\medskip\n\n\\noindent \\emph{Proof of claim.} Consider the short exact sequence of towers $1 \\rightarrow \\{\\ker(f_k)\\} \\rightarrow \\{G_k\\} \\rightarrow \\{H_k\\} \\rightarrow 1$\nand the associated long exact sequence of \\cref{lem:les}. Since (a) the map $\\lim_k f_k \\colon \\lim_k G_k \\to \\lim_k H_k$ is an isomorphism, (b) $\\{G_k\\}$ is Mittag--Leffler, and (c) \\cref{lem:ML-no-lim1}, it follows that $\\lim_k \\ker(f_k)$ and $\\lim^1_k \\ker(f_k) $ both vanish. Invoking \\cref{lem:McGibbon-Moller}, we see that $\\{\\ker(f_k)\\}$ is Mittag--Leffler, so by the first part of the proof $\\{\\ker(f_k)\\}$ is pro-isomorphic to $\\{c\\,0\\}$. The result follows since a levelwise surjective map of tower of groups is a pro-isomorphism if its towers of levelwise kernels are pro-isomorphic to $\\{c\\,0\\}$ \\cite[Proposition III.2.2]{BousfieldKan}.\\end{proof}\n\nMittag--Leffler towers often behave well with $T$-localisation in the sense of \\cref{sec:localisation-groups}:\n\n\\begin{lem}\\label{lem:lim-loc} Let $\\{G_k\\}$ be Mittag--Leffler. If $\\lim_k G_k$ is countable, then the canonical map \n\\[\n\t\\textstyle{\\big({\\lim}_k\\, G_k\\big)_T \\longrightarrow \\lim_k\\big((G_k)_T)}\n\\] is an isomorphism for any set of primes $T$.\\end{lem}\n\n\\begin{proof}By \\cref{lem:dydak-segal} the canonical map of towers $\\{c\\, \\lim_kG_k\\} \\rightarrow \\{G_k\\}$ is a pro-isomorphism, as $\\lim_kG_k$ is countable. As $(-)_T$ preserves pro-isomorphisms and limits of constant towers, the canonical map from the constant tower on $\\{\\lim_kG_k\\}_T$ to $\\{(G_k)_T\\}$ is a pro-isomorphism and the result follows from \\cref{lem:pro-iso-lim}.\\end{proof}\n\n\\begin{rem}We stated \\cref{lem:lim-loc} in terms of $T$-localisation since this is what we will use, but the same proof applies to $(-)_T$ replaces by any endofunctor on the category of groups.\n\\end{rem}\n\n\\subsubsection{Towers of spaces}\nGiven a tower $X_0 \\leftarrow X_1 \\leftarrow \\cdots$ of based spaces, taking homotopy groups results in a tower of pointed sets $\\{\\pi_i(X_k)\\}$ (of groups for $i \\geq 1$). The limits of these towers fit into the following \\emph{Milnor exact sequence} \\cite[Theorem IX.3.1]{BousfieldKan}.\n\n\\begin{lem}\\label{lem:milnor-sequence}\nFor a tower $X_0 \\leftarrow X_1 \\leftarrow \\cdots$ of based spaces and $i\\ge0$, there is a natural short exact sequence of pointed sets (of groups for $i\\ge1$)\n\\[\n\t\\textstyle{0 \\rightarrow \\lim^1_k\\pi_{i+1}(X_k)\\rightarrow \\pi_i(\\ensuremath{\\mathrm{holim}}_kX_k)\\rightarrow\\lim_k\\pi_i(X_k)\\longrightarrow 0}.\n\\]\n\\end{lem}\n\nTogether with \\cref{lem:McGibbon-Moller}, this has the following consequence.\n\n\\begin{prop}\\label{prop:holim-tower}\n\tFix $i\\ge1$. For a tower of based spaces $X_0 \\leftarrow X_1 \\leftarrow \\cdots$ such that $\\pi_i(X_k)$ is countable for all $k\\ge0$, at least one of the following statements holds:\n\t\\begin{enumerate}[(i)]\n\t\t\\item \\label{enum:holim-tower-i-1-uncountable} \\label{enum:holim-tower-i-uncountable} $\\pi_*(\\ensuremath{\\mathrm{holim}}_k X_k)$  is uncountable in degree  $i-1$ or $i$,\n\t\t\\item \\label{enum:holim-tower-rat-iso} $({\\lim}_k \\pi_i(X_k))_T \\rightarrow {\\lim}_k(\\pi_i(X_k)_T)$ is an isomorphism for all sets of primes $T$.\n\t\\end{enumerate}\n\tMoreover, if $\\pi_{i+1}(X_k)$ is countable for all $k\\ge0$, then at least one of the following is the case:\n\t\\begin{enumerate}[(i')]\t\n\t\t\\item \\label{enum:holim-tower-i-uncountable-prime} $\\pi_i(\\ensuremath{\\mathrm{holim}}_k X_k)$ is uncountable,\n\t\t\\item \\label{enum-holim-tower-rat-iso-prime} the natural surjection $\\pi_i(\\ensuremath{\\mathrm{holim}}_k X_k) \\rightarrow {\\lim}_k \\pi_i(X_k)$ is an isomorphism.\n\t\\end{enumerate}\n\t\\end{prop}\n\n\\begin{proof}By \\cref{lem:McGibbon-Moller}, the assumption that $\\pi_i(X_k)$ is countable for $k\\ge0$ implies that either (a) $\\lim^1_k(\\pi_i(X_k))$ is uncountable or (b) the tower is Mittag--Leffler. In case (a), we apply \\cref{lem:milnor-sequence} in degree $i-1$ to conclude that $\\pi_{i-1}(\\ensuremath{\\mathrm{holim}}_k X_k)$ is uncountable, so \\ref{enum:holim-tower-i-1-uncountable} holds. In case (b), either $\\pi_i(\\ensuremath{\\mathrm{holim}}_k X_k)$ is uncountable and \\ref{enum:holim-tower-i-uncountable} holds, or it is countable and \\cref{lem:milnor-sequence} in degree $i$ implies that ${\\lim_k}\\, \\pi_{i}(X_k)$ is countable, so \\ref{enum:holim-tower-rat-iso} holds by \\cref{lem:lim-loc}. Similarly if $\\pi_{i+1}(X_k)$ is countable  for $k\\ge0$, then either $\\pi_i(\\ensuremath{\\mathrm{holim}}_k X_k)$ is uncountable and \\ref{enum:holim-tower-i-uncountable-prime} holds, or this group is countable and so $\\lim_k^1\\pi_{i+1}(X_k)$ is countable by \\cref{lem:milnor-sequence} and thus vanishes by \\cref{lem:McGibbon-Moller}, so the claim follows from the Milnor exact sequence.\n\\end{proof}\n\n\\subsection{Applications to maps between operads}\nTogether with \\cref{thm:truncated-operad-maps}, we use \\cref{prop:holim-tower} to prove the following result about the map\n\\[\n\t\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})\\longrightarrow \\mathrm{Map}^h(\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}})\n\\]\nfor $1$-reduced dendroidal Segal spaces $\\ensuremath{\\catsingle{O}}$ and $\\ensuremath{\\catsingle{P}}$ in the sense of \\cref{section:operads}.\n\n\\begin{thm}\\label{thm:uncountability-for-general-operads}Let $\\ensuremath{\\catsingle{O}}$ and $\\ensuremath{\\catsingle{P}}$ be $1$-reduced dendroidal Segal spaces such that for all $k\\ge0$\n\\begin{itemize}\n\\item all components of $\\ensuremath{\\catsingle{P}}(k)$ are nilpotent and have finitely generated homotopy groups,\n\\item $\\ensuremath{\\catsingle{O}}(k)$ and $\\mathrm{Latch}_k(\\ensuremath{\\catsingle{O}})_{h\\Sigma_k}$ are weakly equivalent to finite CW complexes,\n\\end{itemize}\nthen for all $i\\ge1$ and all basepoints $f\\in\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})$, at least one of the following is the case:\n\t\\begin{enumerate}\n\t\t\\item $\\pi_{*}(\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}))$ is uncountable in degree $i-1$ or $i$,\n\t\t\\item the canonical map $\\pi_{i}(\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}))_\\ensuremath{\\mathbf{Q}} \\to \\pi_i(\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}}))$\n\t\tis an isomorphism.\n\t\\end{enumerate}\n\\end{thm}\n\\begin{proof}During the proof, we implicitly use the equivalence $\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}})\\simeq \\mathrm{Map}^h(\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}})$ and its truncated analogue (see \\cref{lem:loc-dendr-map}). Then \\eqref{equ:goppl-convergence} gives\n\\[\n\t\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})\\simeq \\ensuremath{\\mathrm{holim}}_k\\,\\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}) \\quad \\text{and} \\quad \\mathrm{Map}^h(\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}})\\simeq \\ensuremath{\\mathrm{holim}}_k\\,\\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}}),\n\\] so from the two Milnor sequences (see \\cref{lem:milnor-sequence}) together with the fact that $\\pi_i(\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}}))$ is $\\ensuremath{\\mathbf{Q}}$-local as the homotopy group of a $\\ensuremath{\\mathbf{Q}}$-local space (see \\cref{lem:loc-dendr-map}), we obtain a square\t\\[\\begin{tikzcd}\n\t\\pi_i(\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}))_\\ensuremath{\\mathbf{Q}}\\rar{\\circled{1}}\\dar & \\big({\\lim}_k\\pi_i(\\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}))\\big)_\\ensuremath{\\mathbf{Q}} \\dar{\\circled{2}}\\\\\n\t\\pi_i(\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}}))\\rar{\\circled{3}} & \\lim_k\\pi_i(\\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}}))\n\\end{tikzcd}\\]\nAssuming that $\\pi_*(\\mathrm{Map}^h(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}))$ is countable in degrees $i-1$ and $i$, we need to show that the left vertical map is an isomorphism, which we do proving that this holds for the three circled maps. By \\cref{thm:truncated-operad-maps} \\ref{enum:truncated-fg}, the homotopy groups of  $\\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}})$ are finitely generated for all $k$, so they are in particular countable. By  \\cref{prop:holim-tower} \\ref{enum-holim-tower-rat-iso-prime}, this implies that  $\\circled{$1$}$ is an isomorphism, even before rationalisation. By \\cref{thm:truncated-operad-maps} \\ref{enum:truncated-loc}, we have $\\pi_{k}(\\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{O}},\\ensuremath{\\catsingle{P}}))_\\ensuremath{\\mathbf{Q}} \\cong \\pi_{k}(\\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}}))$ for all $k\\ge1$, so  $\\circled{$2$}$ is an isomorphism by \\cref{prop:holim-tower} \\ref{enum:holim-tower-rat-iso}. Finally, by the Milnor sequence (see \\cref{lem:milnor-sequence}), $\\circled{$3$}$ is surjective and its kernel agrees with $\\lim^1_k\\pi_{i+1}(\\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}}))$, so is an isomorphism if $\\{\\pi_{i+1}(\\mathrm{Map}^h_{\\le k}(\\ensuremath{\\catsingle{P}}_\\ensuremath{\\mathbf{Q}},\\ensuremath{\\catsingle{O}}_\\ensuremath{\\mathbf{Q}}))\\}$ is Mittag--Leffler. For this it suffices that it is a tower of finite-dimensional vector spaces, which is indeed the case by \\cref{thm:truncated-operad-maps} \\ref{enum:truncated-loc} and \\ref{enum:truncated-fg}.\\end{proof}\n\n\\subsubsection{Applications to maps between $E_n$-operads}\\label{sec:maps-between-En-operads}\nHere and henceforth, we write $E_n$ for any operad weakly equivalent to the operad of little $n$-discs (the unital version, so $E_n(0)\\simeq*$). We consider $E_n$ via its dendroidal nerve as a dendroidal Segal space, denoted by the same symbol, and abbreviate its $T$-localisation (see \\cref{sec:localisation-dendroidal-spaces}) by $E_n^T$. Since the spaces of $0$- and $1$-operations of $E_n$ are weakly contractible, \\cref{prop:comparison-of-models} says that the derived mapping spaces between $E_n$-operads do not depend on whether we consider them as operads or dendroidal Segal spaces. Keeping this in mind, we use Theorems~\\ref{thm:truncated-operad-maps} and~\\ref{thm:uncountability-for-general-operads} to prove the following two results:\n\n\\begin{thm}\\label{thm:truncated-ed}Fix $n\\ge1$ and $m\\ge 3$, and a set of primes $T$. For $f\\in\\mathrm{Map}^h_{\\le r}(E_n,E_{m})$ and $k\\ge0$, the following holds:\n\\begin{enumerate}\n\\item the path component $\\mathrm{Map}^h_{\\le k}(E_n,E_{m})_f$ is nilpotent,\n\\item the map $(r_T\\circ(-))\\colon \\mathrm{Map}^h_{\\le k}(E_n,E_{m})_f\\rightarrow \\mathrm{Map}^h_{\\le k}(E_n,E_{m}^T)_{r_T\\circ f}$\nis a $T$-localisation,\n\\item the homotopy groups of $\\mathrm{Map}^h_{\\le k}(E_n,E_{m})_f$ are finitely generated.\n\\end{enumerate}\n\\end{thm}\n\nThe case of \\cref{thm:truncated-ed} that will be relevant to the proof of \\cref{bigthm:nontrivial} and \\cref{bigcor:top-vs-auted} in the next section is $n=m$. For $m-n\\ge2$ and $(-)_T$ being rationalisation, this result appears also in Section 10.2 of \\cite{FTW} (see Remark 10.9 and Proposition 10.10 loc.\\.cit.).\n\n\\begin{thm}\\label{thm:haut-uncountable-or-iso} Fix $n\\ge1$ and $m\\ge 3$. For all $i\\ge1$ and any basepoint in $\\mathrm{Map}^h(E_n,E_m)$, at least one of the following statements holds:\n\\begin{enumerate}\n\t\\item \\label{enum:haut-u-o-i-unc} $\\pi_{*}(\\mathrm{Map}^h(E_n,E_m))$ is uncountable in degrees $i-1$ or $i$,\n\t\\item \\label{enum:haut-u-o-i-iso} the canonical map $\\pi_{i}(\\mathrm{Map}^h(E_n,E_m))_\\ensuremath{\\mathbf{Q}} \\to \\pi_{i}(\\mathrm{Map}^h(E_n^\\ensuremath{\\mathbf{Q}},E^\\ensuremath{\\mathbf{Q}}_m))$\n\tis an isomorphism.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}[Proof of Theorems~\\ref{thm:truncated-ed} and \\ref{thm:haut-uncountable-or-iso}]\nThis follows from Theorems~\\ref{thm:truncated-operad-maps} and ~\\ref{thm:uncountability-for-general-operads} once we checked the hypothesis. The space of $k$-ary operations $E_n(k)$ is weakly equivalent to the space of ordered configurations $F_k(\\ensuremath{\\mathbf{R}}^n)$, so $E_n$ is $1$-reduced for all $n\\ge1$. Moreover, by transversality $E_n(k)$ is $1$-connected (so in particular nilpotent) as long as $n\\ge3$, so its homotopy groups are finitely generated if its homology groups are. We are thus left to show that $E_n(k)_{h\\Sigma_k}$ and $\\mathrm{Latch}_k(E_n)_{h\\Sigma_k}$ have the weak homotopy type of finite CW complexes for all $n\\ge1$ and that $E_n(k)$ has degreewise finitely generated homology groups for $n\\ge3$. By \\cite[Examples 1.1.6, 2.1.13]{Goppl} (see also \\cite[Proposition 3.4.6]{WeissDalian}), the map $\\mathrm{Latch}_n(E_n)\\rightarrow E_n(k)$ agrees up to weak equivalence of $\\Sigma_k$-spaces with the boundary inclusion $\\partial \\mathrm{FM}_n(k)\\subset \\mathrm{FM}_n(k)$ of the Fulton--MacPherson compactification of $F_k(\\ensuremath{\\mathbf{R}}^n)$. This is a compact manifold with corners and free $\\Sigma_k$-action \\cite{Sinha}, so we conclude (i) that $(E_n(k))_{h\\Sigma_k}\\simeq FM_n\/\\Sigma_k$ and $(\\mathrm{Latch}_k(E_n))_{h\\Sigma_k}\\simeq \\partial FM_n\/\\Sigma_k$ have the weak homotopy type of smooth compact manifolds with corners so are weakly equivalent to finite CW complexes, and (ii) that $E_n(k)\\simeq \\mathrm{FM}_n(k)$ has degreewise finitely generated homology.\n\\end{proof}\n\n\\section{\\cref{bigthm:nontrivial}: nontriviality}\\label{sec:nontrivial}\nIn this section we prove results on the homotopy groups of the homotopy fibre $\\mathrm{Aut}^h(E_d)\/\\mathrm{Top}(d)$ of the map $\\mathrm{BTop}(d)\\rightarrow\\mathrm{BAut}^h(E_d)$ mentioned as \\eqref{equ:topd-to-ed-intro} in the introduction (and explained below), and deduce results on the homotopy groups of $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^d)$; \\cref{bigthm:nontrivial} and \\cref{bigcor:top-vs-auted} will follow as special cases. As explained in the outline in \\cref{sec:intr-nontriviality}, the main ingredient besides \\cref{thm:haut-uncountable-or-iso} is work of Boavida de Brito--Weiss' \\cite{BdBWConf} and work of Fresse--Turchin--Willwacher \\cite{FTW}. We also make use of results of Krannich, Kupers, Randal-Williams, and Watanabe \\cite{KrRW,K-RWdiscs,WatanabeII}, though this can be avoided in most cases (see \\cref{rem:classical-proof}).\n\n\n\\subsection{A theorem of Boavida de Brito--Weiss}\\label{sec:conf-cats} \nWe first extract the relevant parts of \\cite{BdBWConf}. By Theorem 1.2 loc.cit., the space $\\mathrm{Map}^h(E_d,E_d) = \\smash{\\mathrm{Map}^h_{\\cat{Opd}}}(E_d,E_d)$ of derived operad maps is equivalent to a mapping space between certain $\\infty$-categories of configurations spaces associated to $\\ensuremath{\\mathbf{R}}^d$. These configuration categories only depend on the underlying \\emph{topological} manifold, so this in particular shows that the standard action of $\\ensuremath{\\mathrm{O}}(d)$ on $E_d$ factors through an action of $\\mathrm{Top}(d)$ and thus gives a map\n\\begin{equation}\\label{equ:top-to-auted-config}\n\t\\mathrm{BTop}(d)\\longrightarrow \\mathrm{BAut}^h(E_d).\n\\end{equation} \nWe will explain below how a reformulation of further results of Boavida de Brito--Weiss relates the homotopy fibre $\\mathrm{Aut}^h(E_d)\/\\mathrm{Top}(d)$ of this map to the $\\ensuremath{\\icat{D}\\mathrm{isc}}$-structure space $S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(D^d)$ of a disc. To state the precise result, we denote by \n\\[\n\t\\Omega^{d+1}_{\\ensuremath{\\mathrm{O}}(d)} \\big({\\mathrm{Aut}^h(E_d)}\/{\\mathrm{Top}(d)}\\big)\\subseteq \\Omega^{d+1}\\big({\\mathrm{Aut}^h(E_d)}\/{\\mathrm{Top}(d)}\\big)\n\\] \nthe collection of those path components that are sent to classes in the image of the map $\\pi_{d+1}(\\mathrm{BO}(d))\\rightarrow \\pi_{d+1}(\\mathrm{BTop}(d))$ under the map\n\\[\n\t\\pi_0(\\Omega^{d+1}({\\mathrm{Aut}^h(E_d)}\/{\\mathrm{Top}(d)}))=\\pi_{d+1}({\\mathrm{Aut}^h(E_d)}\/{\\mathrm{Top}(d)})\\longrightarrow\\pi_{d+1}(\\mathrm{BTop}(d)).\n\\]\n\n\\begin{thm}[Boavida de Brito--Weiss]\\label{thm:sdisc-auted-topd}\nFor $d \\neq 4$ there exists a $0$-coconnected map of the form \n\\[\n\t\\Omega^{d+1}_{\\ensuremath{\\mathrm{O}}(d)} ({\\mathrm{Aut}^h(E_d)}\/{\\mathrm{Top}(d)})\\longrightarrow S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^d).\n\\]\n\\end{thm} \n\nRecall that being $0$-coconnected basically means being an ``inclusion of path components'', meaning a map that induces an injection on $\\pi_0(-)$ and an isomorphism on $\\pi_i(-)$ for $i\\ge1$.\n\n\\begin{proof}[Proof of \\cref{thm:sdisc-auted-topd}]This can be deduced from \\cite{BdBWConf} as follows: combining their Theorems 1.2 and 1.4 with their Section 6 (see also Equation (1.3)), there is contractible space $\\ensuremath{\\mathrm{C}}(D^d,D^d)$ (a certain mapping space of configuration categories) which fits into a homotopy cartesian square\n\\[\\begin{tikzcd} T_\\infty \\ensuremath{\\mathrm{Emb}}_\\partial(D^d,D^d) \\rar \\dar & \\ensuremath{\\mathrm{C}}(D^d,D^d) \\dar \\\\[-2pt]\n\t\\mathrm{Bun}_\\partial(TD^d,TD^d) \\rar & \\Omega^d \\mathrm{Map}^h(E_d,E_d) \\end{tikzcd} \\]\nwhere $\\mathrm{Bun}_\\partial(D^d,D^d)$ is the space of vector bundle maps of $TD^d$ that are fixed on the boundary and $T_\\infty \\ensuremath{\\mathrm{Emb}}_\\partial(D^d,D^d)$ is the embedding calculus approximation to $\\ensuremath{\\mathrm{Emb}}_\\partial(D^d,D^d)$. The bottom horizontal map admits a factorisation \n\\begin{equation}\n\t\\label{equ:factorisation-looped}\\mathrm{Bun}_\\partial(TD^d,TD^d)\\rightarrow \\mathrm{Bun}_\\partial(TD^d,TD^d)^{\\mathrm{Top}}\\rightarrow \\Omega^d \\mathrm{Map}^h(E_d,E_d)\n\\end{equation} \nthrough the space of topological microbundle maps (compare the proof of Theorem 1.6 loc.cit.). Under the equivalences $\\mathrm{Bun}_\\partial(TD^d,TD^d)\\simeq \\Omega^d\\mathrm{O}(d)$ and $\\mathrm{Bun}_\\partial(TD^d,TD^d)^{\\mathrm{Top}}\\simeq \\Omega^d\\mathrm{Top}(d)$, this agrees with the $d$-fold looping of the composition $\\ensuremath{\\mathrm{O}}(d)\\rightarrow\\mathrm{Top}(d)\\rightarrow\\mathrm{Aut}^h(E_d)$. \n\nNow the factorisation \\eqref{equ:factorisation-looped} allows us to form the commutative diagram\n\\[\\begin{tikzcd}[row sep=0.3cm, column sep=0.3cm] \n\t\\ensuremath{\\mathrm{Emb}}_\\partial(D^d,D^d) \\arrow{dd}\\arrow{rr} \\arrow{rd} &[-15pt] &[-15pt] \\ensuremath{\\mathrm{Emb}}_\\partial(D^d,D^d)^{\\mathrm{Top}} \\arrow{dd} \\arrow{rd} &[-15pt] \\\\[-5pt]\n\t& T_\\infty \\ensuremath{\\mathrm{Emb}}_\\partial(D^d,D^d) \\arrow[crossing over]{rr} & & \\ensuremath{\\mathrm{C}}(D^d,D^d) \\arrow{dd} \\\\[-5pt]\n\t\\mathrm{Bun}_\\partial(TD^d,TD^d) \\arrow{rr} \\arrow[equal]{rd} & & \\mathrm{Bun}_\\partial(TD^d,TD^d)^{\\mathrm{Top}}\\arrow{rd} & \\\\[-5pt]\n\t& \\mathrm{Bun}_\\partial(TD^d,TD^d) \\arrow[from=uu,crossing over] \\arrow{rr} & & \\Omega^n \\mathrm{Map}^h(E_d,E_d)\n\\end{tikzcd}\\]\nwhose front and back face are homotopy cartesian; the former by what was we above and the latter by smoothing theory (see \\cite[Essay V]{KirbySiebenmann}; this uses $d\\neq 4$). Note that $\\ensuremath{\\mathrm{Emb}}_\\partial(D^d,D^d) = \\ensuremath{\\mathrm{Diff}}_\\partial(D^d)$ and $\\ensuremath{\\mathrm{Emb}}_\\partial(D^d,D^d)^\\mathrm{Top} = \\ensuremath{\\mathrm{Homeo}}_\\partial(D^d)$. From the construction in \\cite{BdBWConf}, one sees that this diagram is in fact a diagram of $A_\\infty$-spaces if one uses the $A_\\infty$-structure on $T_\\infty \\ensuremath{\\mathrm{Emb}}_\\partial(D^d,D^d)$ by composition induced by the model for embedding calculus from \\cite{BdBWSheaf} which agrees with the $A_\\infty$-structure provided by our model as a result of \\cref{prop:comparison-to-pedromichael} (see the discussion at the beginning of \\cref{sec:comparison-to-pedromichael}). Using contractibility of $\\ensuremath{\\mathrm{C}}(D^d,D^d)$ and of $\\ensuremath{\\mathrm{Homeo}}_\\partial(D^d)$ (using the Alexander trick) and \\cref{thm:emb-calc}, the diagram becomes a map of homotopy fibre sequences\n\\[\\begin{tikzcd}\n\t\\ensuremath{\\mathrm{Diff}}_\\partial(D^d) \\dar{E}\\rar&\\Omega^d\\ensuremath{\\mathrm{O}}(d)\\rar\\arrow[d,equal]&\\Omega^d\\mathrm{Top}(d)\\dar\\\\[-2pt]\n\t\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{\\partial D^d\\times I}}}(E_{D^d},E_{D^d}) \\rar&\\Omega^d\\ensuremath{\\mathrm{O}}(d)\\rar&\\Omega^d\\mathrm{Aut}^h(E_d).\n\\end{tikzcd}\\]\nof $A_\\infty$-spaces. Here we used the abbreviation from \\eqref{equ:abbreviate-right-modules}. Aside from the bottom left fibre, all spaces in this diagram are visibly group-like, so this fibre is as well, i.e.\n\\[\\mathrm{Map}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{\\partial D^d\\times I}}}(E_{D^d},E_{D^d})=\\mathrm{Aut}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{\\partial D^d\\times I}}}(E_{D^d}),\\]\nusing the notation from \\cref{sec:disc-structure-spaces}. We may thus deloop the diagram once (after restricting the components of the rightmost spaces to those in the image of the maps from $\\Omega^d\\ensuremath{\\mathrm{O}}(d)$) and take vertical homotopy fibres to get\n\\[\n\t\\Omega^{d+1}_{\\ensuremath{\\mathrm{O}}(d)}\\mathrm{Aut}(E_d)\/\\mathrm{Top}(d) \\simeq \\mathrm{Aut}_{\\ensuremath{\\icat{M}\\mathrm{od}}(d)_{E_{\\partial D^d\\times I}}}(E_{D^d})\/\\ensuremath{\\mathrm{Diff}}_\\partial(D^d).\n\\]\nThe right-hand space is a collection of components of $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^d)$ by \\eqref{equ:disjoint-union-description-sdisc}, so the claim follows.\n\\end{proof}\n\n\\subsection{Some results of Fresse--Turchin--Willwacher}Next, we recall part of work of Fresse--Turchin--Willwacher \\cite{FTW}, who gave a complete description of the rational homotopy groups of $\\mathrm{Map}^h(E_n,\\smash{E_{m}^\\ensuremath{\\mathbf{Q}}})$ in terms of certain \\emph{graph complexes}. We collect the parts of their results that are relevant to us below, after explaining why they are applicable in our setting.\n\n\\subsubsection{A comparison}The derived mapping spaces $\\mathrm{Map}^h(E_n,E^\\ensuremath{\\mathbf{Q}}_{m})$ considered in \\cite{FTW} differ a priori from those we considered in \\cref{sec:maps-between-En-operads} in two ways:\n\n\\subsubsection*{$\\mathrm{(i)}$}\nFirstly, the derived mapping spaces between operads considered in \\cite{FTW} are formed not in the usual category $\\mathrm{s}\\mathrm{Op}$ of simplicial operads as we did in \\cref{section:operads}, but instead in a certain category $\\mathrm{s}\\Lambda\\mathrm{Op}_{\\varnothing*}$ of connected simplicial $\\Lambda$-operads, equipped with levelwise weak equivalences. This category is isomorphic to the full subcategory $\\mathrm{s}\\mathrm{Op}_{*1}\\subset \\mathrm{s}\\mathrm{Op}$ of the category of simplicial operads such that $\\cat{P}(0)$ and $\\cat{P}(1)$ are singletons (see the discussion following Proposition 4.4. loc.cit.). The inclusion functor $\\mathrm{s}\\Lambda\\mathrm{Op}_{\\varnothing*}\\cong \\mathrm{s}\\mathrm{Op}_{*1}\\to \\mathrm{s}\\mathrm{Op}$ induces weak equivalences on derived mapping spaces: by \\cite{FTWSub} the inclusion $\\mathrm{s}\\mathrm{Op}_{*} \\to \\mathrm{s}\\mathrm{Op}$ induces weak equivalences on derived mapping spaces, and the same holds for $\\mathrm{s}\\mathrm{Op}_{*1} \\to \\mathrm{s}\\mathrm{Op}_*$ since this full subcategory inclusion preserves fibrant and cofibrant objects in suitable model categories on these categories with levelwise weak equivalences (see \\cite[p.~369]{Fresse2} where this is explained in terms of the isomorphic categories $\\mathrm{s}\\Lambda\\mathrm{Op}_{\\varnothing*}\\subset \\mathrm{s}\\Lambda\\mathrm{Op}_{\\varnothing}$). Hence the mapping spaces $\\mathrm{Map}^h(E_n,E_m)$ considered in \\cite{FTW} agree with any of the variants of mapping space we discussed in \\eqref{equ:different-models-mapping-spaces} as a result of \\cref{prop:comparison-of-models}, using that the space of $0$- and $1$-operations of $E_n$ are weakly contractible.\n\n\\subsubsection*{$\\mathrm{(ii)}$}\nSecondly, the authors in \\cite{FTW} rationalise operads differently than we do, namely via a rationalisation functor of Fresse \\cite[Section 12.2]{Fresse2} (therein denoted $\\smash{LG_\\bullet\\Omega_\\sharp^*}(-)$ and phrased in terms of the isomorphism $\\mathrm{s}\\Lambda\\mathrm{Op}_{\\varnothing*}\\cong\\mathrm{s}\\mathrm{Op}_{*1}$ mentioned above) that we denote\n\\[\n\t(-)_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}}\\colon \\mathrm{s}\\mathrm{Op}_{*1}\\longrightarrow \\mathrm{s}\\mathrm{Op}_{*1}.\n\\]\nThis functor comes with a natural transformation $r_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}}\\colon \\mathrm{id}\\rightarrow (-)_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}}$ and has the property that any operad $\\cat{P}\\in \\mathrm{s}\\mathrm{Op}_{*1}$, the induced maps $r_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}}\\colon \\cat{P}(k)\\rightarrow \\cat{P}_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}}(k)$ agree up to weak equivalence with the Sullivan rationalisation as long as $\\ensuremath{\\mathrm{H}}^*(\\cat{P}(k);\\ensuremath{\\mathbf{Q}})$ is degreewise finite dimensional for all $k\\ge1$ (see Theorem 2.2.1 loc.cit.). We can compare this to the rationalisation $(-)_\\ensuremath{\\mathbf{Q}}$ we use (that is, levelwise $T$-localisation for $T$ the set of all primes) as follows:\n\n\\begin{lem}If $\\cat{P}\\in \\mathrm{s}\\mathrm{Op}_{*1}$ is levelwise nilpotent such that $\\ensuremath{\\mathrm{H}}^*(\\cat{P}(k);\\ensuremath{\\mathbf{Q}})$ is degreewise finite dimensional for all $k\\ge1$, then there exists a natural zig-zag of weak equivalences \n\\[\n\tN_d(\\cat{P}_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}})\\simeq (N_d(\\cat{P}))_{\\ensuremath{\\mathbf{Q}}}\n\\] \nbetween the dendroidal nerve of Fresse's rationalisation and the rationalisation of the dendroidal nerve in the sense of \\cref{section:localisation-dendroidal-spaces}.\n\\end{lem}\n\n\\begin{proof}Consider the zigzag $N_d(\\cat{P}_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}})\\xlra{r_\\ensuremath{\\mathbf{Q}}} (N_d (\\cat{P}_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}}))_\\ensuremath{\\mathbf{Q}} \\xleftarrow{N_d(r_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}})_\\ensuremath{\\mathbf{Q}}} (N_d (\\cat{P}))_\\ensuremath{\\mathbf{Q}}$. To check both these maps are weak equivalences, it suffice to do so levelwise. Using the dendroidal Segal condition, we may verify this on corollas. For those, the zig-zag becomes\n\\[\n\t\\cat{P}_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}}(k)\\xlra{r_\\ensuremath{\\mathbf{Q}}} \\cat{P}_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}}(k)_\\ensuremath{\\mathbf{Q}} \\xlla{r_{\\mathrm{F}\\ensuremath{\\mathbf{Q}}}}\\cat{P}(k)_\\ensuremath{\\mathbf{Q}}.\n\\]\nUnder the assumption on $\\cat{P}(k)$, Sullivan rationalisation agrees with the rationalisation in \\cref{sec:localisation}, so all three spaces in the zig-zag are $\\ensuremath{\\mathbf{Q}}$-local and the two maps are weak equivalences.\n\\end{proof}\n\n\\subsubsection{Homotopy groups of spaces of maps between rationalised $E_n$-operads}\\label{sec:FTW}\nThe ingredient from \\cite{FTW} required for the proofs of \\cref{bigthm:nontrivial} and \\cref{bigcor:top-vs-auted} is a computation of the homotopy groups of $\\mathrm{Map}^h(E_d,E^\\ensuremath{\\mathbf{Q}}_d)$ based at the rationalisation map $r_\\ensuremath{\\mathbf{Q}}\\colon E_d\\rightarrow E_d^\\ensuremath{\\mathbf{Q}}$ in a range of degrees, which we summarise as the first two items in the following theorem. In its statement, we write $\\ensuremath{\\mathbf{Q}}[k]$ for the $\\ensuremath{\\mathbf{Z}}$-graded $1$-dimensional vector space concentrated in degree $k$ and we write $\\iota\\colon E_d\\rightarrow E_{d+k}$ for the standard inclusion.\n\n\n\\begin{thm}[Fresse--Turchin--Willwacher]\\label{thm:homotopy-Aut-En}\\\n\t\\begin{enumerate}\n\t\t\\item \\label{enum:fwt-list-even} For $2n\\ge4$, we have an inclusion of graded rational vector spaces\n\t\t\\begin{align*}\n\t\t\t\\pi_{*>0}(\\mathrm{Map}^h(E_{2n},E^\\ensuremath{\\mathbf{Q}}_{2n}),r_\\ensuremath{\\mathbf{Q}})\\supset&\\textstyle{\\Big(\\bigoplus_{i\\ge0}\\ensuremath{\\mathbf{Q}}[2n-4i-1]\\Big)}\\oplus\\\\\n\t\t\t&\\ensuremath{\\mathbf{Q}}[6n-6]\\oplus\\ensuremath{\\mathbf{Q}}[10n-10]\\oplus\\ensuremath{\\mathbf{Q}}[12n-15]\\oplus\\\\\n\t\t\t&\\ensuremath{\\mathbf{Q}}[14n-14]\\oplus \\ensuremath{\\mathbf{Q}}[16n-16]\\oplus \\ensuremath{\\mathbf{Q}}[16n-19]\\oplus  \\\\\n\t\t\t& \\ensuremath{\\mathbf{Q}}[18n-18]\\oplus\\ensuremath{\\mathbf{Q}}[18n-21].\n\t\t\\end{align*}\n\t\t This inclusion is an equality in degrees $*\\le20n-28$.\n\t\t\\item \\label{enum:fwt-list-odd} For $2n+1\\ge3$, we have an inclusion of graded rational vector spaces\n\t\t\\begin{align*}\n\t\t\t\\pi_{*>0}(\\mathrm{Map}^h(E_{2n+1},E^\\ensuremath{\\mathbf{Q}}_{2n+1}),r_\\ensuremath{\\mathbf{Q}})\\supset&\\textstyle{\\Big(\\bigoplus_{i\\ge0}\\ensuremath{\\mathbf{Q}}[2n-4i-2]\\Big)}\\oplus\\\\\n\t\t\t&\\ensuremath{\\mathbf{Q}}[4n-1]\\oplus\\ensuremath{\\mathbf{Q}}[6n-3]\\oplus\\ensuremath{\\mathbf{Q}}[8n-5]\\oplus\\\\\n\t\t\t&\\ensuremath{\\mathbf{Q}}^2[10n-7]\\oplus\\ensuremath{\\mathbf{Q}}^2[12n-9]\\oplus \\ensuremath{\\mathbf{Q}}[12n-6]\\oplus \\\\\n\t\t\t& \\ensuremath{\\mathbf{Q}}^3[14n-11]\\oplus\\ensuremath{\\mathbf{Q}}[14n-8].\n\t\t\\end{align*}\n\t\t This inclusion is an equality in degrees $*\\le16n-14$.\n\t\t\t\\item \\label{enum:fwt-cerf-lemma}$((-)\\circ\\iota)\\colon \\mathrm{Map}^h(E_d,E^\\ensuremath{\\mathbf{Q}}_{d})_{r_\\ensuremath{\\mathbf{Q}}}\\rightarrow \\mathrm{Map}^h(E_{d-1},E^\\ensuremath{\\mathbf{Q}}_{d})_{r_\\ensuremath{\\mathbf{Q}}\\circ \\iota}$ is a weak equivalence for $d\\ge2$.\n\t\\item \\label{enum:homotopy-codim-2}$\\pi_{d+1}(\\mathrm{Map}^h(E_d,E^\\ensuremath{\\mathbf{Q}}_{d+2}),r_\\ensuremath{\\mathbf{Q}} \\circ \\iota)$ is an infinite-dimensional $\\ensuremath{\\mathbf{Q}}$-vector space for $d\\ge1$.\n\t\\end{enumerate}\n\\end{thm}\n\\begin{proof}By \\cite[Corollary 5]{FTW}, there is an isomorphism of graded vector spaces of the form $\\pi_{*>0}(\\mathrm{Map}^h(E_{d},E^\\ensuremath{\\mathbf{Q}}_{d}),r_\\ensuremath{\\mathbf{Q}})\\cong H_{*>0}(\\mathrm{GC}_d^2)$ for $d\\ge3$ where $\\mathrm{GC}_d^2$ is a certain graph complex introduced by Kontsevich (see loc.cit.\\,for details). This complex splits into subcomplexes according to the number of loops of the graphs. The subspaces in \\ref{enum:fwt-list-even} and \\ref{enum:fwt-list-odd} are the homologies of the subcomplexes of loop order $\\le 9$ and $\\le 7$ depending on the parity of $d$ (see Equation (4) loc.cit.). The fact that this subspace spans the full homology in the claimed ranges appears as Corollary 6 loc.cit., which proves \\ref{enum:fwt-list-even} and \\ref{enum:fwt-list-odd}. Part \\ref{enum:fwt-cerf-lemma} is Equation (12) loc.cit. Part \\ref{enum:homotopy-codim-2} follows from the isomorphism $\\pi_k(\\mathrm{Map}^h(E_d,E^\\ensuremath{\\mathbf{Q}}_{d+2}),r_\\ensuremath{\\mathbf{Q}} \\circ \\mathrm{inc})\\cong H_{k-1}(\\mathrm{HCG}_{d,d+2})$ of Corollary 3 loc.cit.\\,by considering the $1$-loop contribution to degree $n$ of $\\mathrm{HCG}_{d,d+2}$ explained in Equation (2) loc.cit.\\,and noting that the graph $H_k$ in that equation has degree $d$ for all $k$.\n\\end{proof}\n\n\\begin{rem}\\label{rem:rationalising-source-does-not-matter}Note that we have $\\mathrm{Map}^h(E_{n},E^\\ensuremath{\\mathbf{Q}}_{m})_{r_\\ensuremath{\\mathbf{Q}}\\circ \\iota}\\simeq \\mathrm{Map}^h(E^\\ensuremath{\\mathbf{Q}}_{n},E^\\ensuremath{\\mathbf{Q}}_{m})_{\\iota^\\ensuremath{\\mathbf{Q}}}$ by \\cref{lem:loc-dendr-map}.\n\\end{rem}\n\n\\subsection{Homotopy groups of $\\mathrm{Aut}^h(E_d)\/\\mathrm{Top}(d)$ }\\label{sec:homotopy-auted-topd}\nWe now state our main technical result on the homotopy groups of the fibre $\\mathrm{Aut}^h(E_d)\/\\mathrm{Top}(d)$ of the map $\\mathrm{BTop}(d)\\rightarrow \\mathrm{BAut}^h(E_d)$ from \\eqref{equ:top-to-auted-config}. We phrase the result in terms of the following statement that we will refer to as $(\\operatorname{\\mathbf{H}}^d_{k,m})$. It depends on a choice of dimension $d\\ge1$ and degrees $k,m\\ge 2$.\n\n\\medskip\n\n\\begin{center} \n\\quad\\quad\\quad\\begin{minipage}{11.5cm}At least one of the following two scenarios is the case:\n\\begin{enumerate}\n\\item $\\pi_*(\\mathrm{Aut}^h(E_d)\/\\mathrm{Top}(d))$ is uncountable in degree $k-2$ or $k-1$, or\n\\item $\\pi_{m}(\\mathrm{Aut}^h(E_d)\/\\mathrm{Top}(d))_\\ensuremath{\\mathbf{Q}}$ is nontrivial.\n\\end{enumerate}\n\\end{minipage} \\hfill $(\\operatorname{\\mathbf{H}}^d_{k,m})$\\end{center}\n\n\\begin{thm}\\label{thm:nontriviality-general}The statement $(\\operatorname{\\mathbf{H}}^d_{k,m})$ holds in the following cases:\n\\begin{enumerate}\n\\item\\label{nontriviality-general:ii} dimension $d=3$ and degrees $k=7$ and $m=6$,\n\\item\\label{nontriviality-general:iii} dimension $d=4$ and degrees $k=4$ and $m=4$,\n\\item\\label{nontriviality-general:iv} dimension $d=2n+1\\ge5$, degrees $k\\le 8n-12$ with $k\\equiv0\\modulo{4}$ and $k\\neq 6n-2$, and $m=k$. For $2n+1=5$, the bound $k\\le 8n-12$ can be weakened to $k\\le 8n-8$,\n\\item\\label{nontriviality-general:v} dimension $d=2n\\ge6$, degrees $2n\\le k\\le 8n-12$ with $k\\equiv0\\modulo{4}$, and $m=k$. If $n$ is odd then the condition $2n\\le k$ can be removed.\n\\end{enumerate}\n\\end{thm}\n\nThis in particular shows that the map $\\mathrm{BO}(d)\\rightarrow \\mathrm{BTop}(d)$ is not a weak equivalence for $d\\ge3$, so proves the first part of \\cref{bigcor:top-vs-auted} in these cases (the second part follows by combining Theorems~\\ref{thm:oo-loop-general} and~\\ref{thm:sdisc-auted-topd}). In the low-dimensional case $d\\le2$, the map is an equivalence which one can see by combining the facts that in these dimensions $\\mathrm{BO}(d)\\rightarrow \\mathrm{BTop}(d)$ and $\\mathrm{BO}(d)\\rightarrow\\mathrm{BAut}^h(E_d)$ are weak equivalences, the first by \\cite[Essay V.\\S 5.0(7)]{KirbySiebenmann} and the latter by work of Horel for $d=2$ \\cite[Theorem 8.5]{Horel} and a folklore result for $d=1$.\n\n\\medskip\n \nTo prepare the proof of \\cref{thm:nontriviality-general}, we extract two results on the homotopy groups of the space $\\mathrm{BTop}(d)$ from the literature. The first says they are countable, and its proof requires the following lemma which is likely known to experts but for which we do not know a reference.\n\n\\begin{lem}\\label{lem:homeo-countable}For a compact topological manifold $M$, possible with boundary, or the interior of such a manifold, the homotopy groups of $\\ensuremath{\\mathrm{BHomeo}}_\\partial(M)$ are countable.\\end{lem}\n\n\\cref{lem:homeo-countable} will be a consequence of the following point-set topological fact. Recall that a topological space is \\emph{second countable} if its topology has a countable basis, and \\emph{locally weakly-contractible} if for every neighbourhood $U$ of a point $p$ there exists a weakly-contractible open neighbourhood $V \\subseteq U$ of $p$.\n\n\\begin{lem}\\label{lem:countable-homotopy-criterion} If $X$ is a locally weakly-contractible second countable space, then the homotopy groups of $X$ based at any basepoint are countable.\\end{lem}\n\n\\begin{proof}Recall (for instance from \\cite[VIII.6.3]{Dugundji}) that every second countable space $X$ is \\emph{Lindel\\\"of}, i.e.\\,every open cover has a countable subcover. For locally weakly contractible $X$, we apply this to the collection of all weakly-contractible open subsets to see that $X$ admits a countable open cover by weakly-contractible subsets. As being a locally weakly-contractible second countable space is preserved by passing to an open subset, the same is true for open subsets of $X$. This allows one to inductively construct an open hypercover $U_\\bullet \\to X$ such that each $U_\\bullet$ has countable many components, each of which is weakly-contractible. Now consider the zigzag\n$X \\leftarrow \\mathrm{hocolim}\\,U_\\bullet \\rightarrow \\mathrm{hocolim}\\,\\pi_0(U_\\bullet)$\nwhose left map is the weak homotopy equivalence of \\cite[Theorem 1.3]{DuggerIsaksen} and whose right map is induced by taking path components, so it is also a weak homotopy equivalence since homotopy colimits take objectwise weak homotopy equivalences to weak homotopy equivalences. Now observe that the right term is equivalent to a countable CW complex, e.g.~using the formula in \\cite[Proposition 3.2]{DuggerIsaksen} exhibiting the homotopy colimit as the geometric realisation of a simplicial set with countable sets of $k$-simplices for all $k$, and hence has countable homotopy groups.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{lem:homeo-countable}] For $M$ compact, restriction to the boundary induces a fibration sequence $\\ensuremath{\\mathrm{Homeo}}_\\partial(M) \\rightarrow \\ensuremath{\\mathrm{Homeo}}(M) \\rightarrow \\ensuremath{\\mathrm{Homeo}}(\\partial M)$ as a result of the existence of collars. Hence it suffices to prove the result for the topological group of homeomorphisms of a compact manifold with boundary or the interior of such a manifold, with no boundary condition. This space is second countable in the compact-open topology \\cite[Proposition 5.4]{GleasonPalais} and locally contractible by \\cite[Theorem 1, Theorem 2]{Cernavskii} (or \\cite[Corollary]{CernavskiiRn} for the case $\\ensuremath{\\mathbf{R}}^d$, which also serves an erratum for the previous reference) or \\cite[Corollary 1.1, Corollary 6.1]{EdwardsKirby} (or \\cite[Theorem 4]{Kirby} for the case $\\ensuremath{\\mathbf{R}}^d$), so the claim follows from \\cref{lem:countable-homotopy-criterion}.\n\\end{proof}\n\nApplying \\cref{lem:homeo-countable} to $\\ensuremath{\\mathbf{R}}^d=\\mathrm{int}(D^d)$ we conclude:\n\n\\begin{cor}\\label{cor:btopd-countable} The homotopy groups of $\\mathrm{BTop}(d)$ are countable.\\end{cor}\n\n\\begin{rem}For $d \\neq 4$, \\cref{cor:btopd-countable} also follows by combining \\cite[Lemma 10, p.\\,188]{MilnorCollectionIV} with \\cite[Essay V.\\S 5.0(1)]{KirbySiebenmann}. The advantage of the proof above is that it applies to $d=4$.\\end{rem}\n\nThe second result on $\\mathrm{BTop}(d)$ we will use follows from works of Krannich, Kupers, Randal-Williams, and Watanabe \\cite{KrRW,K-RWdiscs,WatanabeII}. It concerns two commutative squares\n\\begin{equation}\\label{equ:topd-to-top}\n\t\\begin{tikzcd}\n\t\t\\mathrm{BO}(2n)\\dar{\\iota}\\rar{(e,\\mathrm{stab})}\\dar&[5pt]K(\\ensuremath{\\mathbf{Q}},2n)\\times \\mathrm{BO}\\dar{\\mathrm{id}\\times\\iota}[swap]{\\simeq_\\ensuremath{\\mathbf{Q}}}&[-10pt]\\mathrm{BO}(2n+1)\\rar{(E,\\mathrm{stab})}\\dar{\\iota}&[5pt] K(\\ensuremath{\\mathbf{Q}},4n)\\times \\mathrm{BO}\\dar{\\mathrm{id}\\times\\iota}[swap]{\\simeq_\\ensuremath{\\mathbf{Q}}}\\\\\n\t\t\\mathrm{BTop}(2n)\\rar{(e,\\mathrm{stab})}&K(\\ensuremath{\\mathbf{Q}},2n)\\times\\mathrm{BTop}&\\mathrm{BTop}(2n+1)\\rar{(E,\\mathrm{stab})}&K(\\ensuremath{\\mathbf{Q}},4n)\\times\\mathrm{BTop}\n\t\\end{tikzcd}\n\\end{equation}\nwhere the vertical arrows are induced by the inclusion $\\ensuremath{\\mathrm{O}}(d)\\subset\\mathrm{Top}(d)$ and the horizontal arrows by the stabilisation map, the Euler class $e\\in\\ensuremath{\\mathrm{H}}^{2n}(\\mathrm{BTop}(2n);\\ensuremath{\\mathbf{Q}})$, and the odd-dimensional analogue of its square $E\\in\\ensuremath{\\mathrm{H}}^{2n+1}(\\mathrm{BTop}(2n+1);\\ensuremath{\\mathbf{Q}})$ (see \\cite[Sections 1.2.2 and 8.1.1]{KrRW} for further information on this class). That the right vertical maps are rational equivalences follows from the finiteness of the groups $\\pi_*(\\mathrm{Top}\/\\mathrm{O})$ \\cite[Essay V.\\S 5.0(5)]{KirbySiebenmann}.\n\t\t\n\t\t\\begin{thm}\\label{thm:topd-surjectivity}\nThe maps induced by the bottom horizontal arrows\n\\[\\pi_k(\\mathrm{BTop}(2n))_\\ensuremath{\\mathbf{Q}}\\rightarrow\\pi_k(K(\\ensuremath{\\mathbf{Q}},2n)\\times \\mathrm{BTop})_\\ensuremath{\\mathbf{Q}}\\quad\\pi_k(\\mathrm{BTop}(2n+1))_\\ensuremath{\\mathbf{Q}}\\rightarrow\\pi_k(K(\\ensuremath{\\mathbf{Q}},4n)\\times\\mathrm{BTop})_\\ensuremath{\\mathbf{Q}}\\]\t\nare surjective in degrees $k\\le 4n-1$ for all $n$, and in degrees $k\\le 8n-12$ as long as $n\\ge3$. Moreover, the right-hand map for $n=2$ is also surjective in degree $4n$.\n\\end{thm}\n\t\n\\begin{proof}In degrees $*\\le 4n-1$ the claimed surjectivity follows from the classical fact that the upper horizontal arrows are rationally surjective in exactly this range. \n\t\nIn order to show the claim for the bottom horizontal map in the left square of \\eqref{equ:topd-to-top} for $n\\ge3$ in the range $*\\le 8n-12$, it thus suffices to show that the map $\\Omega^{2n}_0\\mathrm{BTop}(2n)\\rightarrow \\Omega^{2n}_0\\mathrm{BTop}$ is surjective on $\\pi_*(-)_\\ensuremath{\\mathbf{Q}}$ for $*\\le 6n-12$ which can be further reduced to showing that the map $\\ensuremath{\\mathrm{BDiff}}_\\partial(D^{2n})\\simeq \\Omega^{2n}_0\\mathrm{Top}(2n)\/\\ensuremath{\\mathrm{O}}(2n)\\rightarrow \\Omega^{2n}_0\\mathrm{Top}\/\\mathrm{O}(2n)$ is surjective on $\\pi_*(-)_\\ensuremath{\\mathbf{Q}}$ for $*\\le 6n-13$; here we have used Morlet's smoothing theory equivalence \\cite[p\\,241]{KirbySiebenmann}. This surjectivity was proved in \\cite[Corollary 6.7]{K-RWdiscs}. By precomposing the map $\\mathrm{BTop}(2n+1)\\rightarrow \\mathrm{BTop}$ with $\\mathrm{BTop}(2n)\\rightarrow\\mathrm{BTop}(2n+1)$, this argument also shows that the bottom horizontal map in the right square of \\eqref{equ:topd-to-top} for $n\\ge3$ is surjective on $\\pi_*(-)_\\ensuremath{\\mathbf{Q}}$ for $*\\le 6n-12$ as long as $*\\neq 4n$.\n\nThis leaves us with showing that for all $n\\ge2$, the bottom horizontal map of the right square of \\eqref{equ:topd-to-top} is surjective on $\\pi_{4n}(-)_\\ensuremath{\\mathbf{Q}}$. Since the pullback of the class $E\\in\\ensuremath{\\mathrm{H}}^{4n}(\\mathrm{BTop}(2n+1);\\ensuremath{\\mathbf{Q}})$ to $\\mathrm{BO}(2n)$ agrees with $e^2$ by definition of $E$ and hence is decomposable, evaluation of the pullback of $E$ on the image of the Hurewicz map $\\pi_{4n}(\\mathrm{BO}(2n))_\\ensuremath{\\mathbf{Q}} \\to H_{4n}(\\mathrm{BO}(2n);\\ensuremath{\\mathbf{Q}})$ is trivial. Hence the fact that the map $\\mathrm{BO}(2n)\\rightarrow \\mathrm{BTop}$ is surjective on $\\pi_{4n}(-)_\\ensuremath{\\mathbf{Q}}$ implies that the direct summand $\\pi_{4n}(\\mathrm{BTop})_\\ensuremath{\\mathbf{Q}}\\subset \\pi_{4n}(K(\\ensuremath{\\mathbf{Q}},4n)\\times \\mathrm{BTop})_\\ensuremath{\\mathbf{Q}}$ is in the image. So we are left with showing that the map $E\\colon \\mathrm{BTop}(2n+1)\\rightarrow K(\\ensuremath{\\mathbf{Q}},4n)$ is nontrivial for all $n\\ge2$. Using the smoothing theory equivalence $\\ensuremath{\\mathrm{BDiff}}^\\mathrm{fr}_\\partial(D^{2n+1})_0\\simeq \\Omega^{2n+1}_0\\mathrm{Top}(2n+1)$ involving the framed diffeomorphism group, the composition \\vspace{-0.8em} \\[\\pi_{4n}(\\ensuremath{\\mathrm{BDiff}}^\\mathrm{fr}_\\partial(D^{2n+1}))_\\ensuremath{\\mathbf{Q}}\\cong \\pi_{4n}(\\mathrm{BTop}(2n+1))_\\ensuremath{\\mathbf{Q}}\\xlra{E}\\ensuremath{\\mathbf{Q}}\\subset\\ensuremath{\\mathbf{R}}\\] agrees by \\cite[Theorem B.4, Remark B.5]{KrRW} up to a constant with the ``Kontsevich class'' $\\zeta_{2,3}$ from \\cite[p.\\,631]{WatanabeII}, so it is nontrivial for $n\\ge2$ by Theorem 3.1 loc.cit and \\cite{WatanabeIIerr}.\\end{proof}\n\n\n\\begin{proof}[Proof of \\cref{thm:nontriviality-general}]\nThroughout the proof, we  use the facts that $\\pi_{k>0}(\\mathrm{BTop})_\\ensuremath{\\mathbf{Q}}$ is $1$-dimensional for $k\\equiv0\\modulo{4}$ and trivial otherwise, and that $\\mathrm{BTop}(d)$ has countable homotopy groups by \\cref{cor:btopd-countable}. We divide the proof into three cases.\n\n\\begin{itemize}[leftmargin=9mm]\n\t\\item[$d=3$]Applying Theorem \\ref{thm:haut-uncountable-or-iso} for $n=m=3$ and $i=6$, we see that either \n\t\\begin{enumerate}[label=(\\alph*)]\n\t\t\\item $\\pi_*(\\mathrm{BAut}^h(E_3))$ is uncountable in degrees $6$ or $7$, or \n\t\t\\item $\\pi_7(\\mathrm{BAut}^h(E_3))_\\ensuremath{\\mathbf{Q}} \\cong \\pi_7(\\mathrm{BAut}^h(E_3^\\ensuremath{\\mathbf{Q}}))$. \n\t\\end{enumerate}\n\tBy the long exact sequence of $\\mathrm{Aut}^h(E_3)\/\\mathrm{Top}(3)\\rightarrow\\mathrm{BTop}(3)\\rightarrow\\mathrm{BAut}^h(E_3)$, there is nothing left to show in the first case since $\\mathrm{BTop}(d)$ has countable homotopy groups. In the second case, we use that firstly the map $\\mathrm{BO}(3)\\rightarrow \\mathrm{BTop}(3)$ is a weak equivalence by \\cite[p.\\,605]{Hatcher} and thus $\\pi_7(\\mathrm{BTop}(3))_\\ensuremath{\\mathbf{Q}} \\cong \\pi_7(\\mathrm{BO}(3))_\\ensuremath{\\mathbf{Q}}$ vanishes, and that secondly \\cref{thm:homotopy-Aut-En} \\ref{enum:fwt-list-odd} combined with \\cref{rem:rationalising-source-does-not-matter} shows that $\\pi_7(\\mathrm{BAut}^h(\\smash{E_3^\\ensuremath{\\mathbf{Q}}}))\\cong \\pi_6(\\mathrm{Map}^h(\\smash{E_3^\\ensuremath{\\mathbf{Q}}},\\smash{E_3^\\ensuremath{\\mathbf{Q}}});\\mathrm{id})$ is nontrivial, in fact at least $3$-dimensional (since $12n-6=14n-8$ for $n=3$). Using the same long exact sequence as before, this shows the claim in the second case.\n\t\t\n\t\\item[$d=4$] The logic is the same as in the case $d=3$: we again apply Theorem \\ref{thm:haut-uncountable-or-iso}, this time for $n=m=4$ and $i=3$, to see that either \n\t\\begin{enumerate}[label=(\\alph*)]\n\t\t\\item $\\pi_*(\\mathrm{BAut}^h(E_4))$ is uncountable in degrees $3$ or $4$, or \n\t\t\\item $\\pi_4(\\mathrm{BAut}^h(E_4))_\\ensuremath{\\mathbf{Q}} \\cong \\pi_4(\\mathrm{BAut}^h(E_4^\\ensuremath{\\mathbf{Q}}))$. \n\t\\end{enumerate}\n\tAs before, there is nothing left to show in the first case. In the second case, we use that firstly $\\pi_4(\\mathrm{BTop}(4))_\\ensuremath{\\mathbf{Q}}$ is at least $2$-dimensional as a result of \\cref{thm:topd-surjectivity} and that secondly $\\pi_4(\\mathrm{BAut}^h({E_4}^\\ensuremath{\\mathbf{Q}}))$ is $1$-dimensional as a result of \\cref{thm:homotopy-Aut-En} \\ref{enum:fwt-list-even} (since $2n-4i-1 = 3$ for $i=0$ and all other terms are in degree $\\geq 7$).\n\t\t\n\t\\item[$d\\ge5$] Theorem \\ref{thm:haut-uncountable-or-iso} for $n=m=d$ and $i=k-1$ shows that either \n\t\\begin{enumerate}[label=(\\alph*)]\n\t\t\\item $\\pi_*(\\mathrm{BAut}^h(E_{d}))$ is uncountable in degrees $k$ or $k-1$, or \n\t\t\\item $\\pi_{k}(\\mathrm{BAut}^h(E_{d}))_\\ensuremath{\\mathbf{Q}} \\cong \\pi_{k}(\\mathrm{BAut}^h(E_{d}^\\ensuremath{\\mathbf{Q}}))$. \n\t\\end{enumerate}\n\tAs previously, nothing is left to show in the first case. In the second case, we first consider odd $d$. If $d=2n+1\\ge 5$ and $1\\le k\\le 8n-12$ (or $1\\le k\\le 8n-8$ if $n=2$) such that $k\\neq 6n-2$ and $k\\equiv0\\modulo{4}$, then we use firstly that $\\pi_{k}(\\mathrm{BTop}(2n+1))_\\ensuremath{\\mathbf{Q}}$ is at least $2$-dimensional if $k=4n$ and otherwise at least $1$-dimensional by \\cref{thm:topd-surjectivity}, and secondly that \\cref{thm:homotopy-Aut-En} \\ref{enum:fwt-list-odd} shows that $\\pi_k(\\mathrm{BAut}^h({E_{2n+1}}^\\ensuremath{\\mathbf{Q}}))$ is trivial for $k\\neq 4n$ and $1$-dimensional for $k=4n$. Finally, for even $d=2n\\ge6$ and $k\\equiv 0\\modulo{4}$ with $2n\\le k\\le 8n-2$ for $n$ even and $k\\le 8n-2$ for $n$ odd, we use a) that $\\pi_{k}(\\mathrm{BTop}(2n))_\\ensuremath{\\mathbf{Q}}$ is at least $1$-dimensional and at least $2$-dimensional for $k=2n$ if $n$ is even by \\cref{thm:topd-surjectivity}, and b) that \\cref{thm:homotopy-Aut-En} \\ref{enum:fwt-list-even} shows that $\\pi_{k}(\\mathrm{BAut}^h({E_{2n}}^\\ensuremath{\\mathbf{Q}}))$ is trivial for $k\\neq 2n$ and $1$-dimensional for $k=2n$. \\qedhere\n\\end{itemize}\n\\end{proof}\n\t\t\n\\begin{rem}\\label{rem:map-doesnt-matter}\nThe proof of \\cref{thm:nontriviality-general} simply compares the homotopy groups of $\\mathrm{Top}(d)$ and $\\mathrm{Aut}^h(E_d)$ abstractly. It does not use anything about the specific map $\\mathrm{Top}(d)\\rightarrow \\mathrm{Aut}^h(E_d)$.\n\\end{rem}\t\t\n\n\n\\subsubsection{Applications to $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^d)$}\nIn view of the $0$-coconnected map of \\cref{thm:sdisc-auted-topd} \\[\\Omega^{d+1}_{\\ensuremath{\\mathrm{O}}(d)}\\mathrm{Aut}^h(E_d)\/\\mathrm{Top}(d)\\rightarrow S_\\partial^{\\ensuremath{\\icat{D}\\mathrm{isc}}}(D^d),\\] as long as $k-d-3\\ge 0$ the statement $(\\operatorname{\\mathbf{H}}^d_{k,m})$ implies the following variant for $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^d)$:\n\n\\medskip\n\n\\begin{center} \n\\quad\\quad\\quad\\begin{minipage}{11cm}At least one of the following two scenarios is the case:\n\\begin{enumerate}\n\t\\item $\\pi_*(S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^d))$ is uncountable in degree $k-d-3$ or $k-d-2$, or\n\t\\item $\\pi_{m-d-1}(S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^d))_\\ensuremath{\\mathbf{Q}}$ is nontrivial.\n\\end{enumerate}\n\\end{minipage} \\hfill $(\\operatorname{\\mathbf{H}}^{d,\\ensuremath{\\icat{D}\\mathrm{isc}}}_{k,m})$\\end{center}\n\n\\medskip\n\n\\noindent For $k-d-3=0$, this implication uses that if $\\pi_{0}(\\Omega^{d+1}\\mathrm{Aut}^h(E_d)\/\\mathrm{Top}(d))$ is uncountable, then so is $\\pi_{0}(\\Omega^{d+1}_{\\ensuremath{\\mathrm{O}}(d)}\\mathrm{Aut}^h(E_d)\/\\mathrm{Top}(d))$. This is because $\\pi_{d+1}(\\mathrm{BTop}(d))$ is countable, so if the domain of the map $\\pi_{d+1}(\\mathrm{Aut}(E_d)\/\\mathrm{Top}(d))\\rightarrow \\pi_{d+1}(\\mathrm{BTop}(d))$ is uncountable, then so is its kernel. Combined with \\cref{thm:nontriviality-general} we therefore obtain:\n\n\\begin{cor}\\label{cor:precise-nontriviality-sdisc}\nUnder the additional assumption $k-d-3\\ge 0$, the statement $(\\operatorname{\\mathbf{H}}^{d,\\ensuremath{\\icat{D}\\mathrm{isc}}}_{k,m})$ holds for all choices of triples $(d,k,m)$ to which  \\cref{thm:nontriviality-general} applies. \n\\end{cor}\n\nWe now use \\cref{cor:precise-nontriviality-sdisc} to prove that $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^d)$ is not contractible for all $d\\ge5$ with $d\\neq3$.\n\t\t\n\\begin{thm}\\label{thm:thm-for-discs}\nFor $d=3$ or $d\\ge5$, the space $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^d)$ is not contractible.\n\\end{thm}\n\n\\begin{proof}\nFor $d=3$, the claim follows from $(\\operatorname{\\mathbf{H}}^{d,\\ensuremath{\\icat{D}\\mathrm{isc}}}_{k,m})$ for the triple $(d,k,m)=(3,7,6)$ since this statement holds true in this case $k-d-3\\ge 0$ and \\cref{thm:nontriviality-general} applies to this triple. In the case $d\\ge5$ the claim follows similarly as long as we ensure that there exists a $k$ such that $k-d-3\\ge0$ and \\cref{thm:nontriviality-general} applies to the triple $(d,k,k)$. For $d=2n+1$ with $n\\ge 4$, we pick the unique $k\\equiv0\\modulo{4}$ with $2n+5\\le k\\le 2n+8$. This satisfies the requirements because $k-d-3\\ge 0$ and $k\\neq 6n-2$ as $2n+8<6n-2$ and $2n+8\\le 8n-12$. For $d=2n+1$ with $n=3$, we choose $k=12$. This works because $k-d-3=2\\ge0$ and $12\\le 8n-12=12$. For $d=2n+1$ with $n=2$, we choose $k=8$ which works using the improvement of the bound since $k-d-3=0\\ge0$ and $k\\le 8n-8=8$. For $d=2n$ with $n\\ge4$, we can pick the unique $k\\equiv0\\modulo{4}$ with $2n+4\\le k\\le 2n+7$ which is valid since $k-d-3\\ge0$ and $k\\le 2n+7\\le 8n-12$. Finally for $d=2n$ with $n=3$ we pick $k=12$ which works because $k-d-3=3\\ge0$ and $k\\le 8n-12=12$.\n\\end{proof}\n\t\t\n\\begin{rem}\\label{rem:classical-proof}\nIf one relaxes the range $k\\le 8n-12$ in \\cref{thm:nontriviality-general} \\ref{nontriviality-general:iv} and \\ref{nontriviality-general:v} to $k\\le 4n-1$, then the proof we gave does not rely on the recent works \\cite{KrRW,K-RWdiscs,WatanabeII}, since the proof of \\cref{thm:topd-surjectivity} does not use them in this range. This is sufficient to deduce \\cref{bigcor:top-vs-auted}. It also gives a weaker version of \\cref{cor:precise-nontriviality-sdisc} that does not rely on these works. The latter is good enough to conclude \\cref{thm:thm-for-discs} \\emph{except in dimensions $d=5,6,7$}.\n\\end{rem}\n\nCombining \\cref{thm:thm-for-discs} with \\cref{cor:homotopy-retract} implies \\cref{bigthm:nontrivial}. \n\n\\begin{rem}\\label{rem:3-manifolds}\nEven though \\cref{thm:thm-for-discs} applies to $d=3$ and all orientable $3$-manifolds $M$ are spin, we cannot conclude that $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ is nontrivial in this case, because our tangential $2$-type invariance result does not apply if $d=3$, so \\cref{cor:homotopy-retract} is not available. Nonetheless, $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ is nontrivial if $M$ embeds into $D^3$ after removing finitely many codimension $0$ discs, since \n\\begin{enumerate}\n\t\\item removing discs does not change the homotopy type of $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ by \\cref{prop:invariance-handles}, \n\t\\item $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^3)$ is a homotopy retract of $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(M)$ if $M$ embeds into $D^3$ by the same argument as in the second part of proof of \\cref{cor:homotopy-retract}, and\n\t\\item $S^{\\ensuremath{\\icat{D}\\mathrm{isc}}}_\\partial(D^3)$ is nontrivial by \\cref{thm:thm-for-discs}. \n\\end{enumerate} This applies in particular to $S^3$ or to the handlebodies $(S^1\\times D^2)^{\\natural g}\\natural(S^2\\times D^1)^{\\natural g}$ for $g,h\\ge0$, with $\\natural$ denoting the boundary connected sum operation.\n\\end{rem}\n\n\\subsection{Positive codimension} We conclude this section with a brief discussion of an analogue of the nontriviality results of the previous section to positive codimension, by which we mean the following: the subgroup $\\ensuremath{\\mathrm{O}}(c)\\subset \\ensuremath{\\mathrm{O}}(d)$ acting on the last $c$ coordinates stabilises the standard inclusion $E_{d-c}\\rightarrow E_d$ for $c\\ge0$ under the $\\ensuremath{\\mathrm{O}}(d)$-action on $\\mathrm{Map}^h(E_{d-c},E_d)$, so we have a map $\\ensuremath{\\mathrm{O}}(d)\/\\ensuremath{\\mathrm{O}}(c)\\rightarrow \\mathrm{Map}^h(E_{d-c},E_d)$. In the same way as in the case $c=d$ discussed in \\cref{sec:conf-cats}, Boavida de Brito--Weiss' work \\cite{BdBWConf} shows that this action factors as a composition\n\\[\n\t\\ensuremath{\\mathrm{O}}(d)\/\\ensuremath{\\mathrm{O}}(c)\\longrightarrow \\mathrm{Top}(d)\/\\mathrm{Top}(d,d-c) \\longrightarrow \\mathrm{Map}^h(E_{d-c},E_d)\n\\]\nwhere $\\mathrm{Top}(d,d-c)\\subset \\mathrm{Top}(d)$ is the subgroup of those homeomorphism that fix $\\{0\\}\\times\\ensuremath{\\mathbf{R}}^{d-c}\\subset \\ensuremath{\\mathbf{R}}^d$. Generalising from the codimension $c=0$ case of \\cref{bigcor:top-vs-auted}, one might wonder whether\n\\begin{equation}\\label{equ:relative-comparison-top-en}\n\t\\mathrm{Top}(d)\/\\mathrm{Top}(d,d-c) \\longrightarrow \\mathrm{Map}^h(E_{d-c},E_{d}),\n\\end{equation}\nis a weak equivalence. In codimension $c\\ge 3$, this was shown to be the case by Boavida de Brito--Weiss \\cite[Theorem 1.6]{BdBWConf} after taking $(d-c+1)$-fold loop spaces. Adapting the methods of the previous subsection, we consider the remaining codimensions $c=1,2$. As before, we phrase the result in terms of the following placeholder statement involving dimension $d\\ge1$, codimension $c\\in\\{1,2\\}$, and degrees $k\\ge3$ and $m\\ge1$.\n\n\\medskip\n\n\\begin{center}\n\\quad\\begin{minipage}{11.5cm}At least one of the following two scenarios is the case:\n\\begin{enumerate}\n\t\\item The homotopy groups \\[\\pi_*\\big(\\mathrm{hofib}_\\iota\\big(\\mathrm{Top}(d)\/\\mathrm{Top}(d,d-c) \\rightarrow \\mathrm{Map}^h(E_{d-c},E_{d})\\big)\\big)\\] are uncountable in degree $k-3$ or $k-2$, or\n\t\\item The homotopy group \\[\\pi_{m-1}\\big(\\mathrm{hofib}_\\iota\\big(\\mathrm{Top}(d)\/\\mathrm{Top}(d,d-c) \\rightarrow \\mathrm{Map}^h(E_{d-c},E_{d})\\big)\\Big)_\\ensuremath{\\mathbf{Q}}\\] is nontrivial.\n\\end{enumerate}\n\\end{minipage} \\hfill $(\\operatorname{\\mathbf{H}}^{d,c}_{k,m})$\\end{center}\n\n\\medskip\n\n\\noindent The proof requires some results about the homotopy groups of the spaces $\\mathrm{Top}(d,d-c)$:\n\n\\begin{lem}\\label{lem:topdd-c} \\quad\n\t\\begin{enumerate}\n\t\t\\item \\label{enum:c-is-1} The map $(-) \\times \\ensuremath{\\mathbf{R}}^{d-1} \\colon \\mathrm{O}(1) \\simeq \\mathrm{Top}(1) \\to \\mathrm{Top}(d,d-1)$ is a homotopy equivalence.\n\t\t\\item \\label{enum:c-is-2} The map $(-) \\times \\ensuremath{\\mathbf{R}}^{d-2} \\colon \\mathrm{O}(2) \\simeq \\mathrm{Top}(2) \\to \\mathrm{Top}(d,d-2)$ is $(d-2)$-connected.\n\t\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nPart \\ref{enum:c-is-1} admits an elementary argument: if $f(-,-) \\colon \\ensuremath{\\mathbf{R}}^{d-1}\\times \\ensuremath{\\mathbf{R}}=\\ensuremath{\\mathbf{R}}^d \\to \\ensuremath{\\mathbf{R}}^d$ is an orientation-preserving homeomorphism fixing $\\ensuremath{\\mathbf{R}}^{d-1}\\times\\{0\\}$ pointwise, then\n\\[\n\t[0,\\infty)\\times \\ensuremath{\\mathbf{R}}^{d-1}\\times \\ensuremath{\\mathbf{R}} \\ni (t,x,s) \\longmapsto f_t(x,s) \\coloneqq \\begin{cases} (x,s) & \\text{if $|s| \\leq t$}, \\\\\n\tf(x,s-t) & \\text{if $s>t$,} \\\\\n\tf(x,s+t) & \\text{if $s<-t$,}\\end{cases}\n\\]\ngives an isotopy of homeomorphisms that extends continuously to $t=\\infty$ with value $\\mathrm{id}_{\\ensuremath{\\mathbf{R}}^d}$ and depends continuously on $f$. If $f$ is orientation-reversing a similar formula works. \n\t\nPart \\ref{enum:c-is-2} is due to Kirby--Siebenmann \\cite[Theorem B]{KirbySiebenmannCodim2} for $d-2\\neq 4$, who deduce it using immersion theory from an existence and uniqueness result for normal bundles of codimension $2$ locally flat embeddings into $d$-manifolds. In the remaining case $d-2=4$, the necessary results on normal bundles of locally flat embeddings of surfaces into $4$-manifolds were established later by Freedman--Quinn \\cite[Section 9.4]{FreedmanQuinn}. \n\\end{proof}\n\n\\begin{rem}\nThe proof of \\cref{cor:btopd-countable} extends to show that $\\mathrm{Top}(d,d-c)$ has countable homotopy groups: use \\cref{lem:countable-homotopy-criterion}, that it is second countable being a subspace of $\\mathrm{Top}(d)$, and that it is locally weakly-contractible by the variant of \\cite[Corollary 7.3]{EdwardsKirby} for this group.\n\\end{rem}\n\n\\begin{thm}\\label{equ:technical-poscodim}The statement $(\\operatorname{\\mathbf{H}}^{d,c}_{k,m})$ holds in the following cases.\n\\begin{enumerate}\n\t\\item\\label{codim1} For $c=1$, it holds for all choices of $(d,k,m)$ to which \\cref{thm:nontriviality-general} applies. \n\t\\item\\label{codim2} For $c=2$, it holds for $d\\ge3$, $k=d$, and $m=d-1$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nFor \\ref{codim1}, we consider the following zig-zag of maps \\vspace{-0.2em}\n\\[\n\t\\mathrm{Top}(d) \\rightarrow \\mathrm{Top}(d)\/\\mathrm{Top}(d,d-1) \\rightarrow \\mathrm{Map}^h(E_{d-1},E_d)\\xrightarrow{r_\\ensuremath{\\mathbf{Q}}\\circ(-)} \\mathrm{Map}^h(E_{d-1},E_d^\\ensuremath{\\mathbf{Q}})\\xleftarrow{(-)\\circ \\iota}\\mathrm{Map}^h(E_{d},E_d^\\ensuremath{\\mathbf{Q}})\n\\]\nAfter taking loop spaces the leftmost and the rightmost arrow become weak equivalences; the former by \\cref{lem:topdd-c} \\ref{enum:c-is-1} and the latter by \\cref{thm:homotopy-Aut-En} \\ref{enum:fwt-cerf-lemma}. Since the homotopy groups of $\\mathrm{Top}(d)$ are countable by \\cref{cor:btopd-countable}, it thus suffices to prove that for choices $k\\ge3$ and $m\\ge1$ as in the claim either\n\\begin{enumerate}[label=(\\alph*)]\n\\item $\\pi_{*}(\\mathrm{Map}(E_d,E_d);\\mathrm{id})$ is uncountable in degrees $k-2$ or $k-1$, or\n\\item the dimension of $\\pi_{m-1}(\\mathrm{Top}(d);\\mathrm{id})_\\ensuremath{\\mathbf{Q}}$ is larger than that of $\\pi_{m-1}(\\mathrm{Map}(E_d,E_d);\\mathrm{id})_\\ensuremath{\\mathbf{Q}}$.\n\\end{enumerate}\nBut we already showed this, as part of the proof of \\cref{thm:nontriviality-general}. To establish \\ref{codim2}, we apply \\cref{thm:haut-uncountable-or-iso} to degree $i=d-1$ to conclude that either \n\\begin{enumerate}[label=(\\alph*)]\n\t\\item $\\pi_*(\\mathrm{Map}^h(E_{d-c},E_{d}),\\iota)$ is uncountable in degrees $d-2$ or $d-1$, or \n\t\\item $\\pi_{d-1}(\\mathrm{Map}^h(E_{d-c},E_{d}),\\iota)_\\ensuremath{\\mathbf{Q}} \\cong \\pi_{d-1}(\\mathrm{Map}^h(E_{d-c},E^\\ensuremath{\\mathbf{Q}}_{d}),r_\\ensuremath{\\mathbf{Q}}\\circ \\iota)$. \n\\end{enumerate}\nSince $\\pi_{d-1}(\\mathrm{Map}^h(E_{d-2},E^\\ensuremath{\\mathbf{Q}}_{d}),r_\\ensuremath{\\mathbf{Q}}\\circ\\iota)$ is infinite-dimensional by \\cref{thm:homotopy-Aut-En} \\ref{enum:homotopy-codim-2}, it suffices to show that the groups $\\pi_*(\\mathrm{Top}(d)\/\\mathrm{Top}(d,d-2))$ are finitely generated in degrees $*\\le d-1$. The latter follows from a combination of the following facts:\n\\begin{enumerate}\n\t\\item $\\pi_*(\\mathrm{Top}(d))$ is finitely generated in degrees $*\\le d-1$ for all $d$.\n\t\\item The map $(-)\\times \\ensuremath{\\mathbf{R}}^{d-2}\\colon \\ensuremath{\\mathrm{O}}(2)\\simeq \\mathrm{Top}(2)\\rightarrow\\mathrm{Top}(d,d-2)$ is $(d-2)$-connected, so in particular $\\pi_*(\\mathrm{Top}(d,d-2))$ is finitely generated in degrees $*\\le d-2$.\n\\end{enumerate}\nThe first statement follows from \\cite[Essay V.\\S 5.0]{KirbySiebenmann} for $d\\neq 4$ and from \\cite[Theorem 8.7A]{FreedmanQuinn} for $d=4$, and the second is \\cref{lem:topdd-c} \\ref{enum:c-is-2}.\n\\end{proof}\n\nUnwrapping the statement, \\cref{equ:technical-poscodim} in particular implies the following:\n\n\\begin{cor}The map \\eqref{equ:relative-comparison-top-en} is not an equivalence if $d \\geq 3$ and $c=\\{1,2\\}$.\\end{cor}\n\n\\begin{rem}There are no maps of the form $E_{d-c} \\to E_d$ for $c<0$. Indeed, by restricting to $2$-ary operations such a map would induce an equivariant map $\\smash{S^{d-c-1}} \\to \\smash{S^{d-1}}$ with respect to the antipodal action, which implies $c\\ge0$ by the Borsuk--Ulam theorem.\n\\end{rem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\nThe idea of extra dimensions made its first appearance in physics by Kaluza and Klein in 1926\\cite{1,2}, who attempted to unify gravitation and electromagnetism using five space-time dimensions. The theory was first suggested by Theodor Kaluza as a higher dimensional theory of gravity, and ultimately led to the eleven-dimensional supergravity theories and the ten-dimensional superstring\\cite{3}. Like Einstein, Kaluza was in quest of what we call \"the unified theory\", that there is a theory that may explain all of the fundamental forces. He endeavored to represent the electromagnetic force in a similar way as gravity in general relativity. The existence of one extra spatial dimension with a couple of special features was suggested. The idea was this: if we want to explain one more force, maybe we need an extra dimension. Therefore, Kaluza imagined that the universe had four instead of three dimensions of space, and accordingly his theory was formulated. Since the extra dimension is not observed, if one day extra dimensions are discovered in our universe, they are probably to be compact along the lines of Kaluza-Klein theory. Several exact solutions of Kaluza-Klein equations have been discovered since the introduction of this theory\\cite{2,4}.\n\nAs it is well-known, Dirac proposed a magnetic monopole with a string singularity extending from the particle's position to infinity\\cite{5}. Afterwards, more magnetic monopole solutions have been developed\\cite{6}. Gross and Perry\\cite{7}, and Sorkin\\cite{8}, simultaneously obtained a special group of solutions of the five-dimensional Kaluza-Klein theory related to a magnetic monopole. Their solutions explained a string singularity if the spatial extra dimension is compactified. Likewise, Gegenberg and Kunstatter introduced another magnetic monopole solution\\cite{9}. According to\\cite{10} the Kaluza-Klein monopole has an important role in M\/String theory. Gross-Perry-Sorkin (GPS) magnetic monopole is a well-known solution of the Kaluza-Klein theory which is a generalization of the self-dual Euclidean Taub-NUT solution\\cite{11}. This procedure can be applied to the other configurations. In a similar way, the Kaluza-Klein magnetic dipole was represented by choosing the Euclidean Kerr solution\\cite{7,21}. \n\nIn the Kaluza-Klein theory, unlike the Yang-Mills theories, there are solitons which are magnetic dipoles. The dipole configuration is made out of monopole, anti-monopole pair. The reason is that a monopole plus anti-monopole has a distinct topology from the vacuum and, moreover, can not classically annihilate. It has been claimed that the Kaluza-Klein dipole\\cite{7} has a property of describing brane\/anti-brane pair. Some of the dipole-like solutions with magnetic flux tubes have been constracted in \\cite{22,23}. In this paper, we consider a Kaluza-Klein string solution of five-dimensional spacetime.\n\nThe structure of this paper is as follows. In section \\ref{sec2}, we briefly bring up the Kaluza-Klein formalism and mention the main properties of the Kaluza-Klein dipole soliton solution. Subsequently, the boosted solution is examined. We will then present a vacuum solution of the Kaluza-Klein theory and investigate its physical properties in section \\ref{sec3}. Section \\ref{sec4} is devoted to summary and discussion.\n\n\n\\section{Kaluza-Klein Theory and the Dipole Soliton Solution}\\label{sec2}\nAs we have already said, in this section we briefly study the Kaluza-Klein theory and then review the Kaluza-Klein dipole soliton solution. Afterwards, the boosted solution is inspected.\n\\subsection{Kaluza-Klein Theory: a brief review}\\label{subsec2-1}\nThe Kaluza-Klein theory postulates the five-dimensional spacetime and the dimensional reduction of the vacuum spacetime\\cite{12}, which leads to four-dimensional gravity coupled to a $U(1)$ Maxwell field and a scalar dilaton field\\cite{13}. It is also assumed that the 5D energy-momentum tensor vanishes and thus\n\\begin{align}\\label{eq1}\n\\hat G_{AB}=0\n\\end{align}\nwhere\n\\begin{align}\\label{eq2}\n\\hat G_{AB}=\\hat R_{AB}-\\frac{1}{2}\\hat R\\hat g_{AB},\n\\end{align}\nis the 5D Einstein tensor. $\\hat R_{AB}$, $\\hat R$ and $\\hat g_{AB}$ are the five-dimensional Ricci tensor, scalar, and metric tensor, respectively. The indices $A, B, ...$ run over $0, 1, 2, 3, 4$, and five-dimensional quantities are denoted by hats\\cite{2}. The equations of motion can be derived by varying the five-dimensional Einstein-Hilbert action\n\\begin{align}\\label{eq3}\nS=-\\frac{1}{16\\pi\\hat G}\\int\\hat R\\sqrt{-\\hat g}\\:d^{4}x\\:dy,\n\\end{align}\nwhere $\\hat G$ is the five-dimensional gravitational constant. The resulting eq.(\\ref{eq2}) can be written in terms of 4D quantities. Consequently, the Kaluza-Klein field equations in four dimensions read\n\\begin{align}\\label{eq4}\n&G_{\\alpha\\beta}=\\frac{\\kappa^{2}\\phi^{2}}{2}T^{EM}_{\\alpha\\beta}-\\frac{1}{\\phi}[\\nabla_{\\alpha}(\\partial_{\\beta}\\phi)-g_{\\alpha\\beta}\\Box\\phi],\\\\\n&\\nabla^{\\alpha}F_{\\alpha\\beta}=-3\\frac{\\partial^{\\alpha}\\phi}{\\phi}F_{\\alpha\\beta},\\\\\n&\\Box\\phi=\\frac{\\kappa^{2}\\phi^{3}}{4}F_{\\alpha\\beta}F^{\\alpha\\beta},\n\\end{align}\nwhere $T^{EM}_{\\alpha\\beta}=\\frac{1}{4}g_{\\alpha\\beta}F_{\\mu\\nu}F^{\\mu\\nu}-F_{\\alpha}\\!^{\\mu}F_{\\beta\\mu}$ is the electromagnetic energy-momentum tensor, and $F_{\\alpha\\beta}=\\partial_{\\alpha}A_{\\beta}-\\partial_{\\beta}A_{\\alpha}$ is the field strength. $\\phi$ and $\\kappa$ are the scalar field and coupling constant for the electromagnetic potential $A_{\\alpha}$, respectively\\cite{14,15}. (Throughout this paper, Greek indices $\\alpha, \\beta, ...$ run over $0, 1, 2, 3$).\n\n\n\\subsection{Dipole Soliton Solution}\\label{subsec2-2}\nThe Kaluza-Klein dipoles are described by the Kerr-Schwarzschild metrics\\cite{16} which can be obtained from the Kerr-Taub-Bolt solutions\\cite{17} which contain both elementry monopoles and elementry dipoles. The Kaluza-Klein dipoles are regular solutions for $3+1$ spatial dimension in which the fourth dimension is periodic. The line element is represented by the following metric\\cite{18}\n\\begin{align}\\label{eq8}\nds^{2}=&-dt^{2}+\\frac{1}{r^{2}-a^{2}\\cos^{2}\\theta}\\big[\\Delta(dy+a\\sin^{2}\\theta d\\psi)^{2}+\\sin^{2}\\theta((r^{2}-a^{2})d\\psi-ady)^{2}\\big]\\nonumber\\\\\n&+(r^{2}-a^{2}\\cos^{2}\\theta)\\big[\\frac{dr^{2}}{\\Delta}+d\\theta^{2}\\big],\n\\end{align}\nwhere\n\\begin{align}\\label{eq9}\n&\\Delta=r^{2}-2mr-a^{2},\\\\\n&\\psi=\\phi+\\Omega y,\n\\end{align}\nand $y$ is identified with period $2\\pi\/\\kappa$. Here\n\\begin{align}\\label{eq9-1}\n\\Omega=\\kappa\\frac{a}{\\sqrt{m^{2}+a^{2}}},\\ \\ \\ \\ \\ \\kappa=\\frac{\\sqrt{m^{2}+a^{2}}}{2m[m+\\sqrt{m^{2}+a^{2}}]}.\n\\end{align}\nThis solution describes a magnetic dipole. When $r\\to\\infty$, the vector potential is given by\n\\begin{align}\\label{eq9-2}\nA_{\\phi}\\sim\\frac{-2ma\\sin^{2}\\theta}{r}.\n\\end{align}\nThe field of the magnetic dipole is pointing along the $z$ axis. This dipole is not produced by rotating currents and it has zero angular momentum. Moreover, the dipole mass is determind by the value of $a$. There is only one fixed point of the Killing vector $\\frac{\\partial}{\\partial y}$, which is at the source of the magnetic dipole, $r=m+\\sqrt{m^{2}+a^{2}}$\\cite{7}.\n\n\\subsubsection{The Boosted Kaluza-Klein Dipole}\\label{subsubsec2-3}\nFrom the five-dimensional standpoint, we implement a boost to the Kaluza-Klein dipole. The proposed boost is along the extra spatial dimension $y$. We determine the boosted coordinates as ($t$, $r$, $\\theta$, $\\phi$, $y$), and consider metric (\\ref{eq8}) with coordinate renamed as ($t^{\\prime}$, $r^{\\prime}$, $\\theta^{\\prime}$, $\\phi^{\\prime}$, $y^{\\prime}$). Accordingly, we apply the following transformations\n\\begin{align}\\label{eq10}\n&t^{\\prime}=t\\cosh\\alpha-y\\sinh\\alpha,\\\\\n&y^{\\prime}=y\\cosh\\alpha-t\\sinh\\alpha,\n\\end{align}\nwhere $\\alpha$ is the boosted parameter. So, the transformed metric is given by\n\\begin{align}\\label{eq11}\nds^{2}=&-\\Big[\\cosh^{2}\\alpha-\\frac{1}{r^{2}-a^{2}\\cos^{2}\\theta}\\big((r^{2}-2mr-a^{2})(1+a\\Omega\\sin^{2}\\theta)^{2}\\sinh^{2}\\alpha\\nonumber\\\\\n&+(r^{2}\\Omega-a^{2}\\Omega-a)^{2}\\sin^{2}\\theta\\sinh^{2}\\alpha\\big)\\Big]dt^{2}+\\big(\\frac{r^{2}-a^{2}\\cos^{2}\\theta}{r^{2}-2mr-a^{2}}\\big)dr^{2}+\\big(r^{2}-a^{2}\\cos^{2}\\theta\\big)d\\theta^{2}\\nonumber\\\\\n&+\\Big[\\frac{(r^{2}-2mr-a^{2})}{r^{2}-a^{2}\\cos^{2}\\theta}a^{2}\\sin^{4}\\theta+\\frac{(r^{2}-a^{2})^{2}\\sin^{2}\\theta}{r^{2}-a^{2}\\cos^{2}\\theta}\\Big]d\\phi^{2}+\\Big[-\\sinh^{2}\\alpha\\nonumber\\\\\n&+\\frac{\\cosh^{2}\\alpha}{r^{2}-a^{2}\\cos^{2}\\theta}\\big((r^{2}-2mr-a^{2})(1+a\\Omega\\sin^{2}\\theta)^{2}+(r^{2}\\Omega-a^{2}\\Omega-a)^{2}\\sin^{2}\\theta\\big)\\Big]dy^{2}\\nonumber\\\\\n&+\\Big[\\frac{1}{r^{2}-a^{2}\\cos^{2}\\theta}\\big((r^{2}-2mr-a^{2})(-2\\cosh\\alpha\\sinh\\alpha)(1+a\\Omega\\sin^{2}\\theta)^{2}\\nonumber\\\\\n&-2(r^{2}\\Omega-a^{2}\\Omega-a)^{2}\\cosh\\alpha\\sinh\\alpha\\sin^{2}\\theta\\big)+2\\cosh\\alpha\\sinh\\alpha\\Big]dtdy\\nonumber\\\\\n&+\\Big[\\frac{1}{r^{2}-a^{2}\\cos^{2}\\theta}\\big(2a^{2}\\sin^{4}\\theta(r^{2}-2mr-a^{2})(1+a\\Omega\\sin^{2}\\theta)\\nonumber\\\\\n&-2(r^{2}-a^{2})(r^{2}\\Omega-a^{2}\\Omega-a)\\sin^{2}\\theta\\cosh\\alpha\\big)\\Big]d\\phi dy+\\Big[\\frac{1}{r^{2}-a^{2}\\cos^{2}\\theta}\\big(-2a^{2}\\sin^{4}\\theta\\sinh\\alpha(r^{2}\\nonumber\\\\\n&-2mr-a^{2})(1+a\\Omega\\sin^{2}\\theta)-(r^{2}\\Omega-a^{2}\\Omega-a)\\sin^{2}\\theta\\sinh\\alpha\\big)\\Big]dtd\\phi\n\\end{align}\nBecause of the $dtd\\phi$ term the metric becomes a stationary rather than static solution. The transformed scalar and gauge fields resulting from the above metric are given by\n\\begin{align}\\label{eq12}\n\\phi^{2}=-\\sinh^{2}\\alpha+\\frac{\\cosh^{2}\\alpha}{r^{2}-a^{2}\\cos^{2}\\theta}\\big[(r^{2}-2mr-a^{2})(1+a\\Omega\\sin^{2}\\theta)^{2}+(r^{2}\\Omega-a^{2}\\Omega-a)^{2}\\sin^{2}\\theta\\big],\n\\end{align}\n\\begin{align}\\label{eq13}\nA_{t}=&\\frac{1}{\\kappa}\\big[\\frac{(r^{2}-2mr-a^{2})(-2\\cosh\\alpha\\sinh\\alpha)(1+a\\Omega\\sin^{2}\\theta)^{2}}{-\\sinh^{2}\\alpha(r^{2}-a^{2}\\cos^{2}\\theta)+\\cosh^{2}\\alpha\\big((r^{2}-2mr-a^{2})(1+a\\Omega\\sin^{2}\\theta)^{2}+(r^{2}\\Omega-a^{2}\\Omega-a)^{2}\\sin^{2}\\theta\\big)}\\nonumber\\\\\n&+\\frac{-2(r^{2}\\Omega-a^{2}\\Omega-a)^{2}\\cosh\\alpha\\sinh\\alpha\\sin^{2}\\theta+2\\cosh\\alpha\\sinh\\alpha}{-\\sinh^{2}\\alpha(r^{2}-a^{2}\\cos^{2}\\theta)+\\cosh^{2}\\alpha\\big((r^{2}-2mr-a^{2})(1+a\\Omega\\sin^{2}\\theta)^{2}+(r^{2}\\Omega-a^{2}\\Omega-a)^{2}\\sin^{2}\\theta\\big)}\\big],\n\\end{align}\n\\begin{align}\\label{eq14}\nA_{\\phi}=&\\frac{1}{\\kappa(r^{2}-a^{2}\\cos^{2}\\theta)}\\times\\nonumber\\\\\n&\\frac{2a^{2}\\sin^{4}\\theta(r^{2}-2mr-a^{2})(1+a\\Omega\\sin^{2}\\theta)-2(r^{2}-a^{2})(r^{2}\\Omega-a^{2}\\Omega-a)\\sin^{2}\\theta\\cosh\\alpha}{-\\sinh^{2}\\alpha(r^{2}-a^{2}\\cos^{2}\\theta)+\\cosh^{2}\\alpha\\big((r^{2}-2mr-a^{2})(1+a\\Omega\\sin^{2}\\theta)^{2}+(r^{2}\\Omega-a^{2}\\Omega-a)^{2}\\sin^{2}\\theta\\big)},\n\\end{align}\nwhich means the boosted Kaluza-Klein dipole also has an electric field. It can be reduced to the early metric (\\ref{eq8}) by setting $\\alpha=0$.\n\n\n\\section{The Solution}\\label{sec3}\nHere, we introduce another metric which is a vacuum five-dimensional solution, having some properties in common with the string solution. The proposed stationary metric is given by\n\\begin{align}\\label{eq19}\nds^{2}=&-\\left(1-C^{2}r^{2}\\sin^{2}\\theta\\right)dt^{2}+\\left(1+I^{2}(r)r^{2}\\sin^{2}\\theta\\right)dr^{2}+r^{2}d\\theta^{2}+r^{2}\\sin^{2}\\theta d\\phi^{2}\\nonumber\\\\\n&+\\left(1+A^{2}r^{2}\\sin^{2}\\theta\\right)dy^{2}+2CI(r)r^{2}\\sin^{2}\\theta dtdr+2Cr^{2}\\sin^{2}\\theta dtd\\phi+2CAr^{2}\\sin^{2}\\theta dtdy\\nonumber\\\\\n&+2I(r)r^{2}\\sin^{2}\\theta drd\\phi+2AI(r)r^{2}\\sin^{2}\\theta drdy+2Ar^{2}\\sin^{2}\\theta d\\phi dy,\n\\end{align}\nwhere the extra spatial coordinate is represented by $y$, and $A$ and $C$ are constants. $I(r)$ is an arbitrary function of $r$. The coordinates are given by $r$, $\\theta$, $\\phi$ with usual ranges $r\\ge0$, $0\\le\\theta\\le\\pi$, $0\\le\\phi\\le2\\pi$ and $0\\le y\\le2\\pi$. The metric has the signature $(-++++)$. We have constructed the metric (\\ref{eq19}) simply by boosting and transforming the $4+1$ flat solution in such a way that it satisfies the vacuum Einstein equations in five dimensions while keeping all metric parameters, although some are obviously superficial.\n\nThe scalar field $\\phi$, and the gauge field $A_{\\mu}$ deduced from the metric (\\ref{eq19}) are\n\\begin{align}\\label{eq20}\n\\phi^{2}=1+A^{2}r^{2}\\sin^{2}\\theta,\n\\end{align}\nand\n\\begin{align}\\label{eq21}\nA_{t}=\\frac{1}{\\kappa}\\frac{CAr^{2}\\sin^{2}\\theta}{\\left(1+A^{2}r^{2}\\sin^{2}\\theta\\right)},\n\\end{align}\n\\begin{align}\\label{eq22}\nA_{r}=\\frac{1}{\\kappa}\\frac{AI(r)r^{2}\\sin^{2}\\theta}{\\left(1+A^{2}r^{2}\\sin^{2}\\theta\\right)},\n\\end{align}\n\\begin{align}\\label{eq23}\nA_{\\phi}=\\frac{1}{\\kappa}\\frac{Ar^{2}\\sin^{2}\\theta}{\\left(1+A^{2}r^{2}\\sin^{2}\\theta\\right)},\n\\end{align}\nrespectively.\n\nThe size of the Kaluza-Klein circle is as follows \n\\begin{align}\\label{eq23-1}\nC_{KK}=\\int_{0}^{2\\pi}\\sqrt{1+A^{2}r^{2}\\sin^{2}\\theta}dy=2\\pi\\sqrt{1+A^{2}r^{2}\\sin^{2}\\theta},\n\\end{align}\nwhich it depends on the values of $A$ and $r$ as well as $\\theta$. Thus, the size of the  Kaluza-Klein circle becomes non-compactified at large $r$ when $A\\neq0$ , and hence the interpretation of the solution as a  Kaluza-Klein breaks down at large distance. Therefore, in this paper we concentrate at small $r$ to satisfy the  Kaluza-Klein reduction.\n\nMoreover, the components of the electromagnetic fields are given by\n\\begin{align}\\label{eq24}\nF_{tr}=-\\frac{1}{\\kappa}\\frac{2ACr\\sin^{2}\\theta}{(1+A^{2}r^{2}\\sin^{2}\\theta)^{2}}=E_{r},\n\\end{align}\n\\begin{align}\\label{eq24-1}\nF_{t\\theta}=-\\frac{1}{\\kappa}\\frac{2ACr^{2}\\sin\\theta\\cos\\theta}{(1+A^{2}r^{2}\\sin^{2}\\theta)^{2}}=rE_{\\theta},\n\\end{align}\n\\begin{align}\\label{eq25}\nF_{\\theta\\phi}=\\frac{1}{\\kappa}\\frac{2Ar^{2}\\sin\\theta\\cos\\theta}{(1+A^{2}r^{2}\\sin^{2}\\theta)^{2}}=-r^{2}\\sin\\theta B_{r},\n\\end{align}\n\\begin{align}\\label{eq26}\nF_{r\\phi}=\\frac{1}{\\kappa}\\frac{2Ar\\sin^{2}\\theta}{(1+A^{2}r^{2}\\sin^{2}\\theta)^{2}}=r\\sin\\theta B_{\\theta},\n\\end{align}\n\\begin{align}\\label{eq27}\nF_{r\\theta}=-\\frac{1}{\\kappa}\\frac{2AI(r)r^{2}\\sin\\theta\\cos\\theta}{(1+A^{2}r^{2}\\sin^{2}\\theta)^{2}}=-rB_{\\phi}.\n\\end{align}\nThe three-dimensional electric and magnetic field lines are shown in Figs. (\\ref{Fig.1}) and (\\ref{Fig.2}), respectively. By converting the magnetic fields from spherical coordinates to a cartesian one and setting $I(r)=0$, we will have a magnetic field along the $z$ axis. An assosiation of the magnetic field components $B_{r}$, $B_{\\theta}$ and $B_{z}$ are given by\n\\begin{align}\\label{eq27-1}\nB_{r}^{2}+B_{\\theta}^{2}=B^{2}_{z}|_{I(r)=0}=\\frac{4A^{2}}{\\kappa^{2}\\phi^{2}}.\n\\end{align}\nWe can easily show that\n\\begin{align}\\label{eq27-2}\n\\overrightarrow{\\nabla}.\\overrightarrow{B}=0\\ .\n\\end{align}\n\\begin{figure}\n\\begin{adjustbox}{center}\n\\begin{tikzpicture}\n\\begin{axis}[xlabel=$x$, ylabel=$y$, zlabel=$z$, view={60}{120},\ndomain=-1:1,\nxmax=1,\nymax=1,\n]\n\\addplot3[cyan,\/pgfplots\/quiver,\nquiver\/u=y,\nquiver\/v=z,\nquiver\/w=x,\nquiver\/scale arrows=0.25,\n-stealth,samples=10] ({-2*x\/(1+x^2+y^2)^2},{-2*y\/(1+x^2+y^2)^2},{0});\n\\end{axis}\n\\end{tikzpicture}\n\\end{adjustbox}\n\\caption{The three-dimensional electric field lines for $A=C=1$}\\label{Fig.1}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{adjustbox}{center}\n\\begin{tikzpicture}\n\\begin{axis}[xlabel=$x$, ylabel=$y$, zlabel=$z$, view={135}{175},\ndomain=-0.3:0.3,\nxmax=0.3,\nymax=0.3,\n]\n\\addplot3[cyan,\/pgfplots\/quiver,\nquiver\/u=y,\nquiver\/v=z,\nquiver\/w=x,\nquiver\/scale arrows=0.015,\n-stealth,samples=10] ({-4*y*(1\/1-(1+x^2+y^2)^1\/2)\/(1+4*x^2+4*y^2)^2*(1+x^2+y^2)^1\/2},{4*x*(1\/1-(1+x^2+y^2)^1\/2)\/(1+4*x^2+4*y^2)^2*(1+x^2+y^2)^1\/2},{-4\/(1+4*x^2+4*y^2)^2});\n\\end{axis}\n\\end{tikzpicture}\n\\end{adjustbox}\n\\caption{The three-dimensional magnetic field lines for $A=1$ and $I(r)=1\/(1-r)$. As we pointed out, $I(r)$ is an arbitrary function of $r$.}\\label{Fig.2}\n\\end{figure}\n\nThe four-dimensional spacetime is described by the following metric which is obtained by performing a Kaluza-Klein reduction\n\\begin{align}\\label{eq28}\nds^{2}_{4D}=&-\\Big(1-\\big(\\frac{C^{2}}{A^{2}}\\big)\\frac{1}{1+\\frac{1}{A^{2}r^{2}\\sin^{2}\\theta}}\\Big)dt^{2}+\\Big(1+\\big(\\frac{I^{2}(r)}{A^{2}}\\big)\\frac{1}{1+\\frac{1}{A^{2}r^{2}\\sin^{2}\\theta}}\\Big)dr^{2}+r^{2}d\\theta^{2}\\nonumber\\\\\n&+\\big(\\frac{1}{A^{2}}\\big)\\frac{1}{1+\\frac{1}{A^{2}r^{2}\\sin^{2}\\theta}}d\\phi^{2}+\\big(\\frac{CI(r)}{A^{2}}\\big)\\frac{1}{1+\\frac{1}{A^{2}r^{2}\\sin^{2}\\theta}}dtdr+\\big(\\frac{C}{A^{2}}\\big)\\frac{1}{1+\\frac{1}{A^{2}r^{2}\\sin^{2}\\theta}}dtd\\phi\\nonumber\\\\\n&+\\big(\\frac{I(r)}{A^{2}}\\big)\\frac{1}{1+\\frac{1}{A^{2}r^{2}\\sin^{2}\\theta}}drd\\phi,\n\\end{align}\nThe inverse metric tensor is given by\n\\begin{align*}\ng^{\\alpha\\beta}=\n\\begin{pmatrix}\n-1 & 0 & 0 & C \\\\\n0 & 1 & 0 & -I(r) \\\\\n0 & 0 & 1\/r^{2} & 0 \\\\\nC & -I(r) & 0 & H(r,\\theta)\n\\end{pmatrix}\n,\n\\end{align*}\nwhere\n\\begin{equation}\\label{eq29-1}\nH(r,\\theta)= \\frac{1}{r^{2}\\sin^{2}\\theta}+A^{2}+I^{2}(r)-C^{2}.\\nonumber\n\\end{equation}\nAccordingly, the Ricci scalar $R$, and the nontrivial quadratic curvature invariant $R^{\\alpha\\beta\\mu\\nu}R_{\\alpha\\beta\\mu\\nu}$, and the curvature singularities are determined\n\\begin{align}\\label{eq29-1}\nr_{1}=&\\frac{1}{A}\\sqrt{\\frac{2(\\cos^{2}\\theta-1)}{\\cos^{4}\\theta-2\\cos^{2}\\theta+1}}\\ ,\\\\\nr_{2}=&-\\frac{1}{A}\\sqrt{\\frac{2(\\cos^{2}\\theta-1)}{\\cos^{4}\\theta-2\\cos^{2}\\theta+1}}\\ ,\\\\\nr_{3}=&\\left(-\\frac{\\cos^{12}\\theta-6\\cos^{10}\\theta+6\\cos^{8}\\theta+16\\cos^{6}\\theta-39\\cos^{4}\\theta+30\\cos^{2}\\theta-8}{\\left(\\cos^{6}\\theta-3\\cos^{4}\\theta+3\\cos^{2}\\theta-1\\right)^{3}}\\right)^{1\/3}\\nonumber\\\\\n&+\\frac{1}{3}\\left(\\frac{3\\cos^{4}\\theta-6\\cos^{2}\\theta+3}{\\cos^{6}\\theta-3\\cos^{4}\\theta+3\\cos^{2}\\theta-1}\\right)\\ .\n\\end{align}\nNote that $r_{2}(\\theta)$ is irrelevant since it is negative, thus not physical. In general, $r_{1}$ is imaginary in $\\theta\\in(0,\\pi)$, and if $\\theta=0,\\pi$ then $r_{1}\\to+\\infty$ which is beyond our near-origin investigation. It is easy to understand that $r_{3}$ is a negative function of coordinate $\\theta$. However, if $\\theta=0, \\pi$, then $r_{3}\\to+\\infty$. Therefore, a string singularity is present.\n\nThe four-dimensional metric has at least two Killing vector $\\partial_{t}$ and $\\partial_{\\phi}$. In order to obtain the Killing horizon, we can use the condition $\\xi^{2}=0$, where $\\xi$ is an otherside timelike Killing vector, therefore we have\n\\begin{align}\\label{eq30}\ng_{\\mu\\nu}\\xi^{\\mu}\\xi^{\\nu}=-1+\\frac{C^{2}r^{2}\\sin^{2}\\theta}{1+A^{2}r^{2}\\sin^{2}\\theta}=0,\n\\end{align}\nwhich gives\n\\begin{align}\\label{eq30}\nr=\\frac{1}{\\sin\\theta}\\frac{1}{\\sqrt{C^{2}-A^{2}}},\n\\end{align}\nthat is similar to the infinite red-shift surface $g_{tt}=0$. Furthermore, the event horizon can be obtained by $g^{rr}=0$. However, for metric (\\ref{eq28}) $g^{rr}=1$, which means there is no event horizon.\n\nAccording to the radial electric and magnetic fields, we can calculate the net electric and magnetic fluxes through any two-dimensional surface. The electric flux can be calculated through\\cite{19}\n\\begin{align}\\label{eq34}\nQ_{E}=-\\int_{\\partial\\Sigma}{d^{n-2}z\\sqrt{|\\gamma^{\\partial\\Sigma}|}n_{\\mu}\\sigma_{\\nu}F^{\\mu\\nu}}\\ ,\n\\end{align}\nwhere $\\Sigma$ is a hypersurface of constant $t$ and $r$ and, $|\\gamma^{\\partial\\Sigma}|$ is the determinant of the induced metric on $\\partial\\Sigma$, $n_{\\mu}$ and $\\sigma_{\\nu}$ are the unit normal vectors are given by\n\\begin{align}\\label{eq34-1}\nn^{\\mu}=(1,0,0,0)\\ \\ ,\\ \\ \\sigma^{\\mu}=(0,1,0,0),\n\\end{align}\ntherefore\n\\begin{align}\\label{eq34-2}\nQ_{E}=-\\lim_{r\\to\\infty}\\int_{s^{2}}r^{2}\\sin(\\theta)d\\theta d\\phi n^{t}\\sigma^{r}F_{tr}.\n\\end{align}\nAfter some calculation the electric flux is shown to be zero\\footnotemark\n\\begin{align}\\label{eq35}\nQ_{E}=0.\n\\end{align}\nThe magnetic flux for metric (\\ref{eq28}) is as follows\n\\begin{align}\\label{eq36}\n\\Phi_{B}=-\\int_{\\partial\\Sigma}{d^{n-2}z\\sqrt{|\\gamma^{\\partial\\Sigma}|}n_{\\mu}\\sigma_{\\nu}\\star F^{\\mu\\nu}}\\ ,\n\\end{align}\nwhere\n\\begin{align}\\label{eq36-1}\n\\star F^{\\mu\\nu}=\\frac{1}{2}\\epsilon^{\\mu\\nu\\rho\\sigma}F_{\\rho\\sigma},\n\\end{align}\nthus\n\\begin{align}\\label{eq37}\n\\Phi_{B}=-\\frac{2\\pi A}{\\kappa}\\lim_{r\\to\\infty}\\int_{0}^{\\pi}\\frac{r^{4}\\sin^{2}\\theta\\cos\\theta}{(1+A^{2}r^{2}\\sin^{2}\\theta)^{2}}d\\theta,\n\\end{align}\nso that the net magnetic flux vanishes, too\\footnotemark[\\value{footnote}]\\footnotetext{Note that the value of the integral becomes zero before we apply $r\\to\\infty$.} \n\\begin{align}\\label{eq39}\n\\Phi_{B}=0.\n\\end{align}\nWe conclude that the solution is not a magnetic monopole.\n\nBecause of the existence of two Killing vectors $\\xi^{\\mu}=\\delta^{\\mu}_{t}$ and $\\xi^{\\mu}=\\delta^{\\mu}_{\\phi}$ we can get the mass $M$ and the angular momentum $J$, which correspond to the time translation and the axial symmetry, respectively. To calculate the conserved quantities we use the following integral\\cite{20}\n\\begin{align}\\label{eq40}\nI=\\frac{1}{8\\pi G}\\int_{s}\\nabla^{n}\\xi^{m}d^{2}\\Sigma_{mn}\\ ,\n\\end{align}\nwhere $d^{2}\\Sigma_{mn}$ is a two-dimensional surface. By using\n\\begin{align}\\label{eq40-1}\nd^{2}\\Sigma_{mn}=\\epsilon_{mn\\theta\\phi}r^{2}\\sin\\theta d\\theta d\\phi,\n\\end{align}\nwe can easily show that the relevant integration measure for the time translation is as follows\n\\begin{align}\\label{eq40-2}\n\\nabla^{n}\\xi^{m}d^{2}\\Sigma_{mn}=2\\nabla ^{r}\\xi^{t}r^{2}\\sin\\theta d\\theta d\\phi.\n\\end{align}\nSubstituting into the integral we obtain\n\\begin{align}\\label{eq41}\nM=\\frac{1}{8\\pi G}\\int 2\\nabla ^{r}\\xi^{t}r^{2}\\sin\\theta d\\theta d\\phi=0.\n\\end{align}\nIn the case of axial symmetry the integral will give $J=0$.\n\n\n\\subsection{Investigation of Special Cases}\\label{subsec3}\nWe now examine the solution for special cases of constant parameters.\n\nTake, for instance, $C=0$ and assuming a well-behaved $I(r)$, we can look for $I(r_{0})=0$, in which $r_{0}$ is a root of the function $I(r)$. In this case, the metric (\\ref{eq19}) becomes static. The electric field becomes zero, the magnetic field is given by Eqs. (\\ref{eq25}) and (\\ref{eq26}), and the scalar dilaton field is (\\ref{eq20}). If $\\theta=0,\\pi$, then $B_{r}\\neq0$ and $B_{\\theta}=0$. If $\\theta=\\frac{\\pi}{2}$, then $B_{r}=0$ and $B_{\\theta}\\neq0$. Consequently, the infinite red-shift surface besomes imaginary and the carvature singularity is a magnetic string singularity. The Kaluza-Klein dipole solution (\\ref{eq8}) \\big[or eq.(30) in \\cite{7}\\big] behaves this case.\n\nIn the $C=0$ case, the electric field is zero and there are three components of the magnetic field in the spherical coordinates. The scalar dilaton field is given by eq. (\\ref{eq20}). Once again, the metric (\\ref{eq19}) becomes static. The event horizon and the infinite red-shift surface do not exist. Taking $\\theta=0,\\frac{\\pi}{2}, \\pi$, $B_{\\phi}$ becomes zero and the other magnetic field components behave like the previous situation.\n\nBy applying $A=0$, we see that there are no electric and magnetic fields and the scalar diaton field would be constant. The event horizon wouldn't exist, therefore the four-dimensional spacetime singularity is naked. The infinite red-shift surface is as follows\n\\begin{align}\\label{eq43}\nr=\\frac{1}{C\\sin^{2}\\theta}.\n\\end{align}\nWe conclude that, there is no magnetic string and the carvature singularity is still a string singularity along $\\theta=0,\\pi$.\n\nBy putting $I(r)=0$, $B_{\\phi}$ becomes zero. Other components of the magnetic and electric fields exist. In this case, the metric (\\ref{eq19}) remains stationary and behaves like the boosted Kaluza-Klein dipole (\\ref{eq11}).\n\n\\section{Conclusion}\\label{sec4}\nWe considered a Kaluza-Klein string solution which included both dipole and boosted dipole soliton solutions for special cases of parameters which appear in the solution. This solution was studied for small $r$ when $A\\neq0$, where the Kaluza-Klein reduction does not break down. It is also not irrelevant to investigate this solution at large $r$ when $A\\neq0$ which leads to the non-compactified Kaluza-Klein approach. The substantial contribution of this paper was the introduction of this solution and investigating the physical properties of the represented solution. The gravitational mass was calculated and shown to vanish. We computed the magnetic charge and demonstrated that the net magnetic flux of the solution would be zero, which means that there is no extended monopole source. The three-dimensional electric and magnetic fields lines were drawn. In general, it was pointed out that the carvature singularity is not covered by a horizon. It was also shown that the infinite red-shift surface is associated with the $A$ and $C$ parameters. As a special case by applying $|C|<|A|$, we figured out that there is no infinite red-shift surface.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro} \nThe transport of elongated particles or microorganisms by a confined flow is\nrelevant to many industrial applications and natural phenomena. \nThis is for instance the case in  bioengineering or  enhanced oil recovery processes or \nin the build-up and structuration of biofilms in flow channels. We are interested in the present work \nin the instabilities of the motion of single elongated particles (here cylinders) free to move in viscous\nflows. The characteristic dimensions of the particles are comparable to those of the section of the flow channels\nso that confinement  influence very strongly their transport and the occurrence of instabilities.\nThis study is therefore relevant for instance, to the transport of \nfibers or long bio-particles in micro-fluidic channels or in porous and fractured media.\n\nPrevious studies in such flow configurations dealt frequently with the prediction or measurement\nof the hydrodynamic forces on static cylinders submitted to a  flow between parallel plates or in \na rectangular channel~\\cite{Richou2004,Richou2005,Semin2009}. \nInvestigations of moving  cylinders in such geometries  were often restricted to stable motions~\\cite{Faxen1946,Dvinsky87a,Dvinsky87b,Eklund1994,Hu95};  studies of flow instabilities in these geometries\ndealt mostly with vortex shedding behind  fixed cylinders \nbetween parallel walls~\\cite{Williamson1996,Camarri2010,Williamson2008}. \nFinally, instabilities occurring during the sedimentation\nof different types of objects in a viscous fluid were mostly analysed when no\nconfinement effects were present~\\cite{Ern2012,Assemat2012}.\nPeriodic fluttering-like motions have also been studied on plates falling in air,\nbut also without considering the effect of confinement~\\cite{Tanabe1994,Belmonte1998,Pesavento2004}.\n\nRegarding  the present configuration, two previous papers  reported experiments and $2D$ numerical simulations of the motion of a tethered~\\cite{Semin2011} or\nfree \\cite{Dangelo2013}  horizontal cylinder of diameter $D$ inside a  parallelepiped Hele-Shaw cell where  \na vertical Poiseuille flow of velocity $U$ is established. \nTransverse oscillations of the cylinder in the aperture $H$ of the cell are observed  both  when the cylinder is tethered and when it can move freely  across as well as in the plane of the cell. \nMoreover, in this latter case, the cylinder displays, in addition, oscillations of its rolling angle about its axis and of its vertical position at frequencies respectively equal to and twice that of the transverse oscillations.  \nAn important feature  is that, in both cases, these oscillations have been observed at  Reynolds numbers $Re $ as low as $20$: this is well below the threshold value generally reported by other authors for vortex shedding in confined geometries \\cite{Sahin2004}. This suggests that one deals with a mechanism different from those associated to the destabilization of the wake of fixed bodies. The instability observed here cannot appear if the cylinder is fixed: it  involves likely a feedback effect\noriginating in the variations of the pressure and viscous forces  induced by the motion of the cylinder. \n\nIn these previous studies the flow was  either  exactly (simulations)~\\cite{Semin2011} or approximately (experiments)~\\cite{Dangelo2013} two dimensional in the whole cell: in the experiments, the length $L$ of the cylinder was indeed nearly equal to the width $W$ of the cell (experiments) and only cases  in which the cylinder remained horizontal were  studied. The present work studies instead  the influence of the length $L$ ($< W$) on the instabilities; it also deals both with oscillations modes transverse to the cell aperture and with fluttering modes in which the cylinder does not remain horizontal.\n\n  After identifying the different flow regimes,  one studies \n the influence of the ratio $L\/W$ on the instability over the range $0.055 \\leq L\/W \\leq 0.94$: as $L\/W$ becomes\n smaller, the influence of the bypass flow between the ends of the \ncylinder and the sides of the cell becomes larger.   Of special interest is the variation of the frequency\n$f$  with the  velocities  $V_{cx}$ and $U$ of the cylinder and the flow and the relation between $V_{cx}$ and $U$. \n Then, the influence of the blockage ratio $D\/H$ is investigated over the range of values:  $0.39 \\leq D\/H \\leq 0.77$. One studies finally the fluttering motion of the cylinder in the plane of the cell which appears at large values of $D\/H$ and\/or $L\/W$: the motion of the cylinder displays then periodic variations of the angle of the cylinder with respect to horizontal and oscillatory  displacements parallel to its axis.\n\n\\section{Experimental setup and procedure}\n\\begin{figure}[htbp]\n\\includegraphics[width=7.7cm]{gianorio_fig1.eps} \n\\caption{Schematic view of the experimental setup.}\n\\label{fig:exp} \n\\end{figure}\nThe experimental setup has been described in detail in ref.~\\onlinecite{Dangelo2013}.\nThe length $L_{cell}$,   width $W$ and aperture  $H$ of the Hele-Shaw cell (Fig.~\\ref{fig:exp}) are\nrespectively   equal  to $290$, $90$ and $2.85\\, \\mathrm{mm}$. \nThe flowing fluid is a water-glycerol solution of concentration in weight  $C = 10 \\%$.\nThe viscosity and density of the solution at $T = 25^o C$ are respectively $\\mu = 1.153\\, \\mathrm{mPa.s}$ and\n$\\rho_f = 1021\\, \\mathrm{kg\/m^3}$.\nThe flow rate varies  between $0$ and $400 \\, \\mathrm{ml\/mn}$ corresponding to mean velocities \n$-25 \\leq U \\leq 0 \\, \\mathrm{mm\/s}$ ($U$ is negative for an upward flow velocity since\nthe vertical axis $U$ is oriented downward). The top part of the cell as a Y-shape so that the local aperture increases\nfrom $2.85$ to $6\\, \\mathrm{mm}$ over a vertical distance of $48\\, \\mathrm{mm}$. \nAll the experiments are performed using plexiglas cylinders of density $\\rho_s = 1.19 \\times 10^3 \\, \\mathrm{kg\/m^3}$.\nTheir lengths range between $5$ and $85 \\, \\mathrm{mm}$ ($0.055 \\le L\/W \\le 0.94$) and their diameter \nbetween $1.1$ and $2.2 \\, \\mathrm{mm}$, ($0.39 \\le D\/H  \\le 0.77$).\n\nAt the beginning of the experiment the cylinders are placed horizontally at the top end of the cell and one lets them drift into the\nconstant aperture region by reducing the flow rate $Q$; $Q$ is then adjusted in order to bring the cylinder at the desired\ninitial location and is kept constant thereafter during the measurements.\n\nThe displacement of the cylinder is monitored by a digital camera viewing the Hele Shaw cell from the front: its resolution is\n $1024 \\times 768$ pixels  and the frame rate  $30\\, {\\rm fps}$.\n In order to analyze the motion of the cylinder, its length is divided into $4$ parts: the two outside ones\n are painted in black while  black staggered stripes parallel to the axis are painted on the  central portions. \nProcessing digitally the images provides first the location of the ends of the cylinder by detecting the ends \nof the outer stripes: from  these data one determines then the coordinates $(x_c, z_c)$ of the center of mass \nof the cylinder and its angle $\\theta$ with respect to the horizontal. \nAs observed previously~\\cite{Dangelo2013}, oscillations of the cylinder transverse to the walls of the \ncell are accompanied by  oscillations at the same frequency $f$ of the angle of rotation \nof the cylinder about its axis: the variations of this angle  are estimated by computing\n the transverse displacement of the staggered stripes painted on the cylinder with respect to those of the ends.\nThe frequency of the oscillations is, here, deduced from the variation of this angle with time.\n\\section{Different cylinder motion regimes}\n\\begin{figure}[hbp]\n\\includegraphics[width=7.7cm]{gianorio_fig2.eps} \n\\caption{Different types of cylinder motions observed as a function of the ratios $D\/H$ and $L\/W$\nfor a plexiglas cylinder and  a water-glycerol solution ($C = 10 \\%$): straight trajectory ($+$);\ntransverse oscillation ($\\square$); fluttering+transverse oscillation ($\\boxtimes$); fluttering ($\\times$).}\n\\label{fig:regime_map} \n\\end{figure}\nThe different types of motion the free cylinder have been identified for different values of the control parameters\n$D$, $L$ and $U$. The diameter and the length of the cylinder were observed to have the largest influence\non the results: we have therefore displayed in Fig.~\\ref{fig:regime_map}  a map of the different regimes \nobserved as a function of the dimensionless parameters $D\/H$ and $L\/W$.\n\\begin{itemize}\n\\item For  $D\/H \\lesssim 0.4$, the cylinder\nfollow a straight stable vertical trajectory with no transverse or side oscillations.\n\\item  For higher ratios $0.4 \\lesssim D\/H \\lesssim 0.6$, it\ndisplays transverse oscillations. When the length $L$ becomes of the order of $W$ ($L\/W \\gtrsim 0.9$), \na fluttering motion   is superimposed onto the transverse oscillations ($D\/H = 0.53$): it corresponds to a periodic variation\nof the angle $\\theta$  with the horizontal with a frequency  significantly lower than that of the transverse\noscillations.\n\\item For $D\/H \\gtrsim 0.6$ a fluttering motion without transverse oscillation generally occurs\nexcept for $L\/W = 0.61$, $D\/H = 0.63$ in which case the two types of oscillations are again superimposed.\n\\end{itemize}\nIn short, increasing the ratio $D\/H$ and, therefore, the transverse confinement results\nin a transition from stable flow to transverse oscillations and then to a fluttering motion:\nmoreover, fluttering appears earlier for strong longitudinal confinements.\n\\section{Influence of the  confinement on the   transverse oscillations}\n\\subsection{Influence of the cylinder length}\n\\begin{figure}[htbp]\n\\includegraphics[width=7.7cm]{gianorio_fig3.eps} \n\\caption{Influence of the length $L$ of plexiglas cylinders of constant diameter $D = 1.5\\, \\mathrm{mm}$ ($D\/H = 0.53$) on\n the variation of  cylinder velocity $V_{cx}$ with the  velocity $U$ of a flow of water  with $10\\%$ glycerol.\n  Symbols:  experimental data; straight lines: linear regressions \nover these data (excluding point $U = 0$). \n$L\/W = 0.055$ ($\\lozenge$), $0.11$ ($+$), $0.22$ ($\\otimes$), $0.33$ ($\\oplus$), $0.44$ ($\\boxplus$), $0.61$ ($\\bigcirc$), $0.67$ ($\\Box$), $0.77$ ($\\times$),  $0.89$ ($\\ast$) and $0.94$ ($\\boxtimes$).\nInset: variation as a function of the cylinder length $L$ of the  slope $\\alpha$ of the linear regressions ($\\triangleright$)\nand of the  velocity $|V_r|$ ($\\triangleleft$).}\n\\label{fig:Vc_U_L} \n\\end{figure}\nThe influence of the lateral confinement parameter $L\/W$ on the transverse \noscillations has first been studied: experiments have been performed   for free  cylinders of  diameter \n$D = 1.5\\, \\mathrm{mm}$ ($D\/H = 0.53$) \nand  $L$ varying between $5$ and $85 \\, \\mathrm{mm}$ ($0.055 \\le L\/W \\le 0.94$).\n\nA first important characteristic is the variation of the velocity $V_{cx}$ of the cylinder as a function \nof that of the flow ($U$)  which is here oriented upward and, therefore, negative. \nThe main graph in Fig.~\\ref{fig:Vc_U_L} shows\nthat $V_{cx}$ varies  linearly with $U$ (data points corresponding to $U = 0$\nare however above the linear trend).\nFor the  curves of  Fig.~\\ref{fig:Vc_U_L} corresponding to $L\/W \\le 0.77$,  only  \ntransverse oscillations occur:  the axis of the cylinder remains horizontal and no flutter is visible. \nFor $L\/W = 0.89$ ($\\ast$) and $L\/W = 0.94$ ($\\boxtimes$), the cylinder both flutters and oscillates transversally. \n\nThe straight lines on the main graph of Fig.~\\ref{fig:Vc_U_L} correspond to  a linear regression on the data\naccording to the equation:\n\n\\begin{equation}\n V_{cx} = \\alpha (U - V_r);\n \\label{eq:Vr}\n \\end{equation}\nThe variations with $L\/W$ of the slope $\\alpha = \\mathrm{d}V_{cx}\/\\mathrm{d}U$ of the regression lines and\n of $V_r$ are plotted in the inset:\n $\\alpha$  depends only weakly  on  $L$, even when fluttering occurs   ($\\triangleright$ symbols) and  \nits values are all in the range $1.4 \\pm 0.1$.\nFrom Eq:\\ref{eq:Vr},   $V_r\\, (< 0)$  is the velocity of the upward flow\nat  which the cylinder remains at a constant average vertical position~\\cite{Dangelo2013};\nmore generally $V_r$ can be considered as a relative velocity of the fluid and the cylinder: the fact\nthat it remains constant for a free cylinder as $V_{cx}$ suggests that the drag force on the \ncylinder is determined by $V_r$ and remains constant as $V_{cx}$ varies for a given free cylinder\nin order to balance its weight.\nIn contrast to  $\\alpha$, $|V_r|$ decreases   as $L$ increases  ($\\triangleleft$ symbols):\n the limit  of $V_r$ as $L \\rightarrow W$  corresponds to the value  $V_{r2D}$ for a $2D$\n configuration with, here: $V_{r2D} = -9\\, \\mathrm{mm.s^{-1}}$. The slope of the variation \n of $|V_r|$ with $L\/W$ is almost constant except at the lowest values for which it increases\n sharply.\n\n\\begin{figure}[htbp]\n\\includegraphics[width=7.7cm]{gianorio_fig4.eps} \n\\caption{Experimental variation of the transverse oscillation frequency $f$ as a function of the\n cylinder velocity $V_{cx}$\nfor plexiglas cylinders of  different dimensionless lengths $L\/W$ and constant dimensionless\ndiameter $D\/H = 0.53$.\nInset : same frequency data as in the main graph plotted as a function of the mean flow velocity $U$. \nIn both graphs, the  symbols are  the same as in Fig.~\\ref{fig:Vc_U_L}.}\n\\label{fig:f_U_L} \n\\end{figure}\nFig.~\\ref{fig:f_U_L}, displays the variation of  the frequency  $f$ with  the cylinder velocity $V_{cx}$ \nfor the different  lengths $L$: one observes then an excellent collapse of the different curves onto a common \nweakly increasing trend, with, for $V_{cx} = 0$, a frequency  $f = 3.3 \\pm 0.1 \\, \\mathrm{Hz}$. The data points\nare much more dispersed  when $f$ is plotted as a function of $U$ (see inset). \n\n\n \n\n\n\n\n\n We explain now    the above result, namely that, for free cylinders,  the frequency $f$ is independent of $L\/W$ \n when the velocity $V_{cx}$ is  kept constant. \n  When  $L \\rightarrow W$ (like in Ref.~\\onlinecite{Dangelo2013}), flow is  \n two dimensional with a zero bypass flow between the ends of the cylinder and the side walls. \n The balance, per unit length, between the weight of the cylinder and the vertical hydrodynamic\nforce  is then~\\cite{Semin2009}:  \n\\begin{equation}\n \\lambda_{p2D}\\, \\mu\\, {U}  - \\lambda_{s2D}\\, \\mu\\, {V}_{cx} = - (\\rho_s - \\rho_f) A\\,  g\n\\label{eq:drag}\n\\end{equation}\n ($A$ is the cylinder section).  Eq.~(\\ref{eq:drag}) can then be rewritten in the form similar\n to Eq.~(\\ref{eq:Vr}):\n\\begin{equation}\n{V}_{cx} = \\frac{\\lambda_{p2D}}{\\lambda_{s2D}} U + \\frac{(\\rho_s - \\rho_f) A g}{\\lambda_{s2D}\\, \\mu} = \\alpha_{2D} (U - V_{r2D}), \\label{eq:Vr_2D}\n\\end{equation} \nin which $V_{r2D}$ and $\\alpha_{2D}$ are constant with $U$.\n$V_{r2D}$ and $\\alpha_{2D}$ will be equal to the limits of $V_r$ and $\\alpha$ when $L\/W \\rightarrow 1$\nwith, for $D\/H = 0.53$, from the inset of Fig.~\\ref{fig:Vc_U_L}: $|V_{r2D}| = 9\\, \\mathrm{mm.s^{-1}}$ and $\\alpha_{2D} = 1.4$\n \nIn the general case  $L < W$, the local flow {in the part of the aperture occupied by}  the cylinder is  still  assumed\n to be  two dimensional: more precisely, one assumes that the velocity component $v_z$ is negligible and that \n $v_x(x,y)$ and $v_y(x,y)$ are independent of $z$ along the length $L$ of the cylinder.\nThe results to be discussed below suggest that this assumption is valid for $L\/D \\gtrsim 7$.\n\nThe balance of forces per unit length on the cylinder should then be  the same as for $L = W$:\nEqs.~\\ref{eq:drag} and ~\\ref{eq:Vr_2D} remain then  valid  with  the  same \nparameters  $\\lambda_{s2D}$ and $\\lambda_{p2D}$  (or $V_{r2D}$ and $\\alpha_{2D}$) \nprovided  $U$ is replaced by  a local velocity $U_{loc}$ constant along the length $L$. Then:\n\\begin{equation}\n{V}_{cx} = \\frac{\\lambda_{p2D}}{\\lambda_{s2D}} U_{loc} + \\frac{(\\rho_s - \\rho_f) A g}{\\lambda_{s2D}\\, \\mu} =\n \\alpha_{2D}(U_{loc} - V_{r2D}). \n\\label{eq:Vr_loc}\n\\end{equation} \n$U_{loc}$ is related to  $V_{cx}$ and $V_{r2D}$ by Eq.~(\\ref{eq:Vr_loc}) and is, therefore, \n independent of $L\/W$. Since the velocities $U_{loc}$ and $V_{cx}$ determine completely\nthe local flow on the  cylinder and, therefore, the frequency $f$, the latter will also  be independent of $L\/W$:\nthis explains the excellent coincidence of the curves of Fig.~\\ref{fig:f_U_L}.\n\nIn order to understand the relation between  $V_r$ and  $L\/W$ displayed in the inset of \nFig.~\\ref{fig:Vc_U_L}, one estimates first the difference $U_{loc} - U$. \nThe flow in each clearance of  width $(L - W)\/2$ between the ends of the cylinder and the sides of the cell (Fig.~\\ref{fig:exp}) is, like  around the cylinder, assumed to be viscous and  two dimensional:   the corresponding velocity \n$U_a$ averaged over the aperture $H$ is then  constant with $z$ along its width $W - L$. \nApplying mass conservation, $U$, $U_a$ and $U_{loc}$  satisfy $U = (U_{loc} \\, L + U_a \\,(W - L))\/W$.\nMoreover, the constant value of $\\alpha$, in particular as $L \\rightarrow W$  allows one to take $\\alpha = \\alpha_{2D}$.\nCombining  these two latter  results with  Eqs.~(\\ref{eq:Vr}) and (\\ref{eq:Vr_loc}) leads then to:\n\\begin{equation}\nV_r - V_{r2D} =  U - U_{loc} = \\frac{W - L}{W} (U_a - U_{loc})\n\\label{Vr_Vr2D}\n\\end{equation}\n\nTaking for simplicity $V_{cx} = 0$, momentum conservation requires that  the force due to the  pressure drop \n$\\Delta p$ between the  upstream and downstream sides of the cylinder balances its effective weight per unit length. Then:\n$\\Delta p\\, H = (\\rho_s - \\rho_f) A  g$ so that $\\Delta p$ is independent of $L\/W$. Assuming a transverse pressure\nequilibrium, the pressure drop   across  the clearance between the cylinder and the walls must also be $\\Delta p$. \nUnder the above assumptions of a  $2D$ viscous flow, the  velocity $U_a$ is proportional\nto $\\Delta p$ with a coefficient independent of the width $W - L$. Like $\\Delta p$, $U_a$ is then constant with $L\/W$ \nand Eq.~(\\ref{Vr_Vr2D}) predicts   the linear variation of $V_r$ with $L\/W$  observed experimentally. \n Still for $V_{cx} = 0$ and $D\/H = 0.53$ one has, from Eq.~(\\ref{eq:Vr_loc}):  \n $U_{loc} = V_{r2D}   = -9\\, \\mathrm{mm.s^{-1}}$ (see above for the determination of $V_{r2D}$). \n Taking $L = 0$ in Eq.~(\\ref{Vr_Vr2D})  leads then to:  $U_a = V_r(L\/W \\rightarrow 0) = -27\\, \\mathrm{mm.s^{-1}}$.\n\n\\subsection{Influence of the  diameter on the transverse oscillations}\\label{influ_D}\n\\begin{figure}[h!]\n\\includegraphics[width=7.7cm]{gianorio_fig5.eps} \n\\caption{Experimental variation of the transverse oscillation frequency as a function of the mean flow velocity $U$\nfor plexiglas cylinders of different diameter to aperture ratios:  $D\/H = 0.46$ ($\\triangledown$, $\\blacktriangledown$), \n$0.53$ ($\\bigcirc$, $\\bullet$), $0.56$ ($\\triangle$, $\\blacktriangle$),  $0.63$ ($\\Join$). \nOpen symbols: $L\/W = 0.61$; black symbols: $L\/W = 0.22$.\nFlowing fluid: water-glycerol solution ($C = 10\\,\\%$). Inset: variation  of the  slope $\\alpha$ and\n the velocity $V_r$ with the diameter $D$. \nData points corresponding to $D\/W = 0.39$ (stable regime), $0.7$ and $0.77$ (pure fluttering regime)  have been\nadded for comparison.}\n\\label{fig:f_D_U} \n\\end{figure}\nThe influence of the transverse confinement parameter $D\/H$ on the transverse oscillations has been investigated\nby  using several  cylinders with different diameters and for two different \nlateral confinements ($L\/W = 0.61$ and $L\/W = 0.22$): the  values of $D\/H$ belonged\nto the interval $0.39 \\le D\/H \\le 0.77$. Transverse oscillations were observed  in the  range $0.46 \\le D\/H \\le 0.63$. \n\nThe inset of Fig.~\\ref{fig:f_D_U} displays the variation of the parameters $\\alpha$ and $V_r$ with $D\/H$: data   points\ncorresponding to  pure  fluttering  ($D\/H = 0.7$ and $0.77$) or to  stable   ($D\/H = 0.39$) regimes have also been\n included in this graph. The experimental value of $\\alpha$ is independent of $D\/H$ with \n  $\\alpha = 1.34 \\pm 0.1$ for both transverse confinement ratios $L_c\/W$;\nmoreover, the transition to the stable or fluttering regimes does not result in any variation of $\\alpha$.\n\nThe velocity $V_r$ decreases smoothly by $30\\%$ as $D\/H$ varies from $0.39$\nto $0.77$ for $L\/W = 0.61$ and increases by $15\\%$ as $D\/H$ varies from $0.39$\nto $0.63$ for $L\/W = 0.22$: like for $\\alpha$, there is no visible influence of the transition from a flow regime\nto another.\n\nThe variation of the  oscillation frequency with the cylinder velocity $V_{cx}$ is plotted in the main\ngraph of Fig.~\\ref{fig:f_D_U} for these same cylinders. All  data points correspond  to pure transverse oscillations  except for $L\/W = 0.61$ and $D\/H = 0.63$ (Fig.~\\ref{fig:regime_map}): in this latter  case, transverse oscillations  and fluttering occur simultaneously.  \nThe frequency $f$ is also  remarkably independent of the ratio $D\/H$ for all values of $D\/H$,\n except for the smallest diameter  $D\/H = 0.46$ and for  $L\/W = 0.61$: in this   particular case, the common trend \n of variation of $f$ with $V_{cx}$ is only followed for  $V_{cx} > 0$. but the values of $f$ are higher for $V_{cx} \\le 0$.\nNo special feature of the variations is observable when fluttering is superimposed onto transverse oscillations.\nThe curves corresponding to the two different values of $L\/W$ ($0.22$ and $0.61$) also coincide which generalizes the results \nobtained for $D\/H = 0.53$ and displayed in Fig.~\\ref{fig:f_U_L}. \n\n\\section{Fluttering oscillations of the cylinder}\n\n\\begin{figure}[htbp]\n\\includegraphics[width=13 cm]{gianorio_fig6.eps} \n\\caption{a) Successive views of the cylinder taken at  time intervals $\\Delta t = 1\/3\\, \\mathrm{s}$ in the fluttering regime. \nb) Variations as a function of time in the same experiment of the geometrical parameters characterizing the\nmotion of the cylinder in the fluttering regime;  $\\delta z_c$: distance from the vertical axis of symmetry of the cell (continuous line), $\\delta x_c$:\ndeviation of the vertical coordinate from a linear variation with time (dotted  line), $\\theta$: angle of the axis with respect to the \nhorizontal (dashed line).  $L\/W = 0.22$, $D\/H = 0.63$, $U = 6.6\\, \\mathrm{mm.s^1}$.}\n\\label{fig:flutter_t} \n\\end{figure}\nThe fluttering instability is characterized by oscillations of the angle $\\theta$ of the axis of the cylinder with respect to\nthe horizontal (Fig.~\\ref{fig:flutter_t}a) and dashed line in Fig.~\\ref{fig:flutter_t}b. These angular oscillations are accompanied by synchronous variations of the lateral displacement $\\delta z_c$ of the center of mass  (continuous line):  the angle $|\\theta|$ reaches an extremal value  shortly after the end of the cylinder is closest to one of the sides of the cell.\n\nThe fluttering motion also induces fluctuations of the vertical velocity $v_x$ of the cylinder. These variations  are \nvisualized in the figure (dotted line) from the deviation $\\delta x_c$ of the vertical coordinate from the linear trend which would \ncorrespond to a constant velocity: $\\delta x_c$ oscillates at twice the fluttering frequency indicating that negative and positive deviations of the angle $\\theta$ have the same influence on the velocity. In the transverse oscillations regime, \nvertical oscillations at a frequency $2f$ have also been observed although, in this case, the cylinder \nremained horizontal~\\cite{Dangelo2013}.\n\n\n\\begin{figure}[htbp]\n\\includegraphics[width=7.7cm]{gianorio_fig7.eps} \n\\caption{Experimental variation of the fluttering frequency $f_f$ for a water-glycerol solution   ($C = 10\\,\\%$)\nas a function of the normalised length $L\/W$ for plexiglas cylinders. Inset at lower left: variation of the  slope $\\alpha$ ($\\vartriangleright$) and   the velocity $V_r$ ($\\vartriangleleft$)  with $L\/W$ for different diameters ($0.53 \\le D\/H \\le 0.77$). \nInset at upper right: variation of $f_f$ with $V_{cx}$ for  cylinders\nwith different values of $D\/H$ and $L\/W$ ($f = 0$ means: no oscillation).\nDimensionless length and diameter  of the cylinders: $D\/H = 0.63$, $L\/W = 0.22$ ($\\Join$); \n$D\/H = 0.77$, $0.22 \\le L\/W \\le 0.61$ ($\\square$);  $D\/H = 0.77$, $L\/W = 0.49$ ($\\#$); $D\/H = 0.53$, $L\/W = 0.89$ ($\\ast$);\n$D\/H = 0.53$, $L\/W = 0.94$ ($\\boxtimes$); $D\/H = 0.63$, $L\/W = 0.61$ ($\\bigcirc$); $D\/H = 0.7$, $L\/W = 0.61$ ($\\oplus$);$D\/H = 0.77$, $L\/W = 0.61$ ($\\otimes$)}\n\\label{fig:ff_Lc_Vr} \n\\end{figure}\nFig.~\\ref{fig:ff_Lc_Vr} displays  variations of the fluttering frequency $f_f$ as a function of the velocity $V_{cx}$ or\nthe lateral confinement $L\/W$ for different pairs of values of the ratios  $L\/W$ and $D\/H$.\nAs mentioned above, the fluttering instability  is  observed  for large values of $D\/H \\ge 0.63$ either \nalone or superimposed onto transverse oscillations (see Fig.~\\ref{fig:regime_map}). For \n$D\/H = 0.53$, fluttering is only observed (together with transverse oscillations) for the largest ratios $L\/W \\ge 0.89$.\n\nA first specific feature of this instability is that the frequency $f_f$ is more than  $3$ times lower than that\n of the transverse oscillations; $f_f$ decreases  significantly with $L\/W$, {\\it e.g.} by a\n factor $3$ as $L\/W$ increases from $0.22$ to $0.9$ (main graph of Fig.~\\ref{fig:ff_Lc_Vr}).\n\nThe strong influence of $L\/W$ on $f_f$ suggests that these oscillations\nare driven by the dissymmetry  between the bypass flows  at the two ends of the cylinders\nwhen it moves laterally  ($\\delta z_c \\neq 0$):\nthe forces at the two ends of the cylinder are then unequal, which creates a torque \n that rotates it and a  lateral force inducing a sideways motion. \n\nFinally, for a given cylinder, $f_f$ is independent of the velocity $V_{cx}$ (and on $U$, too) \nas can be seen in the upper inset of Fig.~\\ref{fig:ff_Lc_Vr}. \nIn this same graph, ones observes that the frequencies  $f_f$ corresponding to the same ratio\n $L\/W = 0.61$ and to different values of $D\/H$ ($0.63$, $0.7$, $0.77$) coincide at all velocities\n  (($\\bigcirc$), ($\\oplus$ and ($\\otimes$) symbols). Similarly, in the main graph, for $L\/W = 0.22$, the\n   values of  $f_f$ corresponding to $D\/H = 0.63$ and $0.77$ are   nearly equal.\n\nRegarding the mean vertical velocity $V_{cx}$, the slope $\\alpha$ of the variation with $U$  is  practically\nindependent of $L\/W$  (insert at lower left of Fig.~\\ref{fig:Vc_U_L}): the common value is the same as that found\npreviously in the stable and transverse oscillation regimes  (insert of Fig.~\\ref{fig:Vc_U_L}) Also, like in the \ncase of transverse oscillations, the velocity  $|V_r|$ decreases significantly as the ratio $L\/W$ increases;\nthe value of $V_r$ is also nearly independent of $D\/H$. \n \n At a first glance, these fluttering instabilities have visual similarities with those observed \n for falling  sheets (or leaves)~\\cite{Tanabe1994,Belmonte1998,Pesavento2004}:  these latter \n experiments are however realized in unconfined configurations. These instabilities, which take \n place in unconfined configurations, take however place at larger Reynolds numbers: they involve \n vortex shedding from the edges of the sheets in contrast with the present ones.\n\\section{Conclusion}\nThe present experiments demonstrate that the motion of a buoyant cylinder in a vertical viscous Hele Shaw cell flow may display \noscillatory transverse and\/or fluttering instabilities depending   on the value of the two\n confinement parameters $L\/W$ and $D\/H$.  \nFor  $0.2 \\le L\/W  \\le 0.8$, for instance,  one shifts continuously  from the  stable regime\n to the transverse oscillations and then to  fluttering as $D\/H$ increases; at the transition, \n the  two oscillatory instabilities  may, in addition, be superimposed. These instabilities are controled by the relative velocity between the fiber and the fluid. They are observed for Reynolds numbers (based on the relative velocity) as low as $20$: the mechanisms of the instabilities are thus different from those associated to destabilization of the wake at the rear of a fiber.\n \n\nIn an approximate description, the  transverse instability  is considered as a  $2D$ one, corresponding \nto a local relative  velocity  with the same value $V_{r2D}$ as for a cylinder of length $L=W$ (the transverse deviations of\nthe flow lines are neglected). $V_{r2D}$ is determined by the cylinder velocity $V_{cx}$ and a local flow velocity $U_{loc}$.\nBoth  $V_{r2D}$ and $U_{loc}$\n cannot be determined directly with the present setup: they can however be assumed to be equal respectively to the \n experimental values of $V_{r}$ and $U$ in the limiting case $L\/W \\rightarrow 1$.\nAs a result, the frequency $f$ is independent of $L\/W$ for a constant velocity $V_{cx}$ (but depends instead \non $L\/W$ for a constant velocity $U$); also, $f$ increases by less than $15  \\%$  when $V_{cx}$ varies from \n$0$ to $20\\, \\mathrm{mm.s^{-1}}$ in agreement with the results report in \\cite{Dangelo2013}.\nThis $2D$  description is not valid for the shortest cylinders ($L\/W = 0.055$) of aspect ratio \n$L\/D = 1.75$. \n\nThe above discussion is only valid for free cylinders.\nFor tethered ones~\\cite{Semin2011} for which $V_{cx} = 0$,\n there is no longer an equilibrium between the hydrodynamic forces on the cylinder and its weight because of\n the tension of the supporting threads.  In this case,  the frequency $f$ depends both on  $U$ \n and on $L\/W$:  $f$ is indeed determined in this case by the local velocity $U_{loc}$. If the fiber and the fluid have the same density (this situation was considered by Berthet~\\cite{Berthet2013}), we have $V_{cx}=\\alpha U$, and a relative velocity $V_r=0$. In this case, fluttering and oscillations in the gap will not be observed.  \n \nReverting to the free case, the frequency  $f(V_{cx})$ is also found experimentally to be  independent \nof the dimensionless diameter  $D\/H$: this result is quite surprising since,  a  first view, several mechanisms\nmight  induce a variation of $f$. Due to the  lack of dependence of $f$ on $L\/W$, one can consider \nthis problem for simplicity in the  $2D$ case equivalent  to  $L\/W =1$.  \nFirst, increasing $D\/H$ increases the section and, therefore, the mass of the cylinder which should reduce\nthe frequency $f$.  Increasing $D$ also \n reduces the clearance $H - D$ and enhances the velocity (and Bernoulli pressure) variations: this should\n instead increase the value of $f$.\n Varying $D$  also influences the relative velocity $V_{r2D}$ and, as a result: $f$. Increasing $D$ \n increases first the weight of the cylinder which is the driving force in Eq.~(\\ref{eq:drag}); it should\n  also increase the drag by reducing the clearance between the cylinder and the front walls. These two \n effects will respectively tend to increase and reduce the relative velocity (and therefore the frequency). $2D$ \n numerical simulations should allow one to determine the relative magnitude of the different effects and\n whether they compensate each other. \n\nIn contrast, the fluttering instability is  strongly related  to  the variations of the distances between \nthe ends of the cylinder and the sides of the cell: understanding it  requires a model at the scale\nof the full width $W$ of the Hele Shaw cell.\n\nIn spite of these differences, the transverse and fluttering instabilities of free cylinders share several \ncommon properties. Both  $f$ and $f_f$ depend weakly, or not at all, on the \n velocity $V_{cx}$ for a given cylinder:  in both cases, this results from the fact \n that, as mentioned above, these frequencies are determined mainly by a relative velocity of the cylinder \n and the fluid: the latter remains constant with the flow velocity, again in order to keep the balance \n between the hydrodynamic  forces and the weight of the cylinder. \n   Also,  $f$ and $f_f$ are both independent of the cylinder diameter: this result involves likely \n   a compensation between different effects and will require $2D$ numerical \n   simulations in  order to be explained. \n  \n\n\nFurther studies are needed to understand better the analogies and  differences of these two types \nof instabilities as well as the weak dependence of variables like $\\alpha$ on all the control parameters \n investigated ($D\/H$, $L\/W$). \nRegarding the fluttering instability, the characteristics of the fluid are important parameters\nto be investigated. \n\\begin{acknowledgments} \nWe thank \nB. Semin for his careful reading of the manuscript and his useful comments and \nJ.E. Wesfreid for useful suggestions. \nWe acknowledge the RTRA Triangle de la Physique\nand the LIA PMF-FMF (Franco-Argentinian International Associated Laboratory\nin the Physics and Mechanics of Fluids). The work of one of us (VD) was supported by\na Bernardo Houssay grant allocated by the Argentinian and French\nministries of research. \n\\end{acknowledgments} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe proximal point method is a conceptually simple   algorithm for minimizing a function $f$ on ${\\mathbb R}^d$.  Given an iterate $x_t$, the method  defines  $x_{t+1}$ to be any minimizer of the proximal subproblem\n$$\\operatornamewithlimits{argmin}_{x}~\\left\\{f(x)+\\tfrac{1}{2\\nu}\\|x-x_t\\|^2\\right\\},$$\nfor an appropriately chosen parameter $\\nu>0$. At first glance, each proximal subproblem seems no easier than minimizing $f$ in the first place. On the contrary, the addition of the quadratic penalty term often regularizes the proximal subproblems and makes them well conditioned. Case in point, the  subproblem may become convex despite $f$ not being convex; and even if $f$ were convex, the subproblem has a larger strong convexity parameter thereby facilitating faster numerical methods. \n\nDespite the improved conditioning, each proximal subproblem still requires invoking an iterative solver. For this reason, the proximal point method has predominantly been thought of as a theoretical\/conceptual algorithm, only guiding algorithm design and analysis rather than being implemented directly. One good example is the proximal bundle method \\cite{bundle}, which approximates each proximal subproblem by a cutting plane model. In the past few years, this viewpoint has undergone a major revision. In a variety of circumstances, the proximal point method (or a close variant) with a judicious choice of the control parameter $\\nu>0$ and an appropriate iterative method for the subproblems can lead to practical and theoretically sound numerical methods. \nIn this article, I will briefly describe three recent examples of this trend: \n\\begin{itemize}\n\t\\item a subgradient method for weakly convex stochastic approximation problems \\cite{prox_guide_subgrad},\n\t\\item  the prox-linear algorithm for minimizing compositions of convex functions and smooth maps \\cite{prox_error,prox_lin_paq,quad_conv,nest_GN,prox,composite_cart},\n\t\\item Catalyst generic acceleration schema \\cite{catalyst} for regularized Empirical Risk Minimization.\n\\end{itemize}\n\n\nIn this article, I will focus only on the proximal point method for minimizing functions, as outlined above. The proximal point methodology applies much more broadly to monotone operator inclusions; I refer the reader to the monograph of Bauschke and Combette \\cite{BC_book} or the seminal work of Rockafellar \\cite{mon_rock}.\n\n\\section{Notation}\nThe following two constructions will play a basic role in the article. For any closed function $f$ on ${\\mathbb R}^d$,   the {\\em Moreau envelope} and the {\\em proximal map} are \n\\begin{align*}\nf_{\\nu}(z)&:=\\inf_{x}~\\left\\{f(x)+\\tfrac{1}{2\\nu}\\|x-z\\|^2\\right\\},\\\\\n{\\textrm prox}_{\\nu f}(z)&:=\\operatornamewithlimits{argmin}_{x}~\\left\\{f(x)+\\tfrac{1}{2\\nu}\\|x-z\\|^2\\right\\},\n\\end{align*}\nrespectively.\nIn this notation, the proximal point method is simply the fixed-point recurrence on the proximal map:\\footnote{To ensure that ${\\textrm prox}_{\\nu f}(\\cdot)$ is nonempty, it suffices to assume that  $f$ is bounded from below.}  $${\\bf Step\\, }t: \\qquad \\textrm{choose }x_{t+1}\\in {\\textrm prox}_{\\nu f}(x_t).$$\n\nClearly, in order to have any hope of solving the proximal subproblems, one must ensure that they are convex. Consequently, the class of weakly convex functions forms the natural setting for the proximal point method. \n\\begin{defn}{\\rm\nA function $f$ is called {\\em $\\rho$-weakly convex} if the assignment $x\\mapsto f(x)+\\frac{\\rho}{2}\\|x\\|^2$ is a convex function.}\n\\end{defn}\n\n For example, a $C^1$-smooth function with $\\rho$-Lipschitz gradient is $\\rho$-weakly convex, while a $C^2$-smooth function $f$ is \n$\\rho$-weakly convex precisely when the minimal eigenvalue of its Hessian is uniformly bounded below by $-\\rho$.\nIn essence, weak convexity precludes functions that have downward kinks. For instance, $f(x):=-\\|x\\|$ is not weakly convex since no addition of a quadratic makes the resulting function convex.\n\n\n Whenever $f$ is $\\rho$-weakly convex and the proximal parameter  $\\nu$ satisfies  $\\nu<\\rho^{-1}$, each proximal subproblem is itself convex and therefore globally tractable. Moreover, in this setting, the Moreau envelope is $C^1$-smooth with the gradient\n\\begin{equation}\\label{eqn:grad_form}\n\\nabla f_{\\nu}(x)=\\nu^{-1}(x-{\\textrm prox}_{\\nu f}(x)).\n\\end{equation}\n Rearranging the gradient formula yields the useful interpretation of the proximal point method as gradient descent on the Moreau envelope\n$$x_{t+1}=x_t-\\nu\\nabla f_{\\nu}(x_t).$$\n\nIn summary, the Moreau envelope $f_{\\nu}$ serves as a $C^1$-smooth approximation of $f$ for all small $\\nu$. Moreover, the two conditions $$\\|\\nabla f_{\\nu}(x_{t})\\|< \\varepsilon$$ and\n$$\\|\\nu^{-1}(x_t-x_{t+1})\\|<\\varepsilon,$$\nare equivalent for the proximal point sequence $\\{x_t\\}$.\n  Hence, the step-size $\\|x_t-x_{t+1}\\|$ of the proximal point method serves as a convenient termination criteria. \n\n\n\n\\subsection{Examples of weakly convex functions}\nWeakly convex functions are widespread in applications and are typically easy to recognize. One common source of weakly convex functions is the composite problem class:\n\\begin{equation}\\label{eqn:comp}\n\\min_{x}~ F(x):=g(x)+h(c(x)),\n\\end{equation}\nwhere $g\\colon {\\mathbb R}^d\\to{\\mathbb R}\\cup\\{+\\infty\\}$ is a closed convex function, $h\\colon{\\mathbb R}^m\\to{\\mathbb R}$ is convex and $L$-Lipschitz, and $c\\colon{\\mathbb R}^d\\to{\\mathbb R}^m$ is a $C^1$-smooth map with $\\beta$-Lipschitz gradient. An easy argument shows that  $F$ is $L\\beta$-weakly convex. This is a worst case estimate. In concrete circumstances, the composite function $F$ may have a much more favorable weak convexity constant (e.g., phase retrieval  \\cite[Section 3.2]{duchi_ruan_PR}).\n\n\n\n\\begin{example}[Additive composite]\\label{exa:add_comp}\n\t{\\rm The most prevalent example is additive composite minimization. In this case, the map $c$ maps to the real line and $h$ is the identity function:\n\t\t\\begin{equation}\\label{eqn:add_comp}\n\t\t\\min_{x}~ c(x)+g(x).\n\t\t\\end{equation}\n\t\tSuch problems appear often in statistical learning and imaging. A variety of specialized algorithms are available; see for example Beck and Teboulle \\cite{smoothing_beckT} or Nesterov \\cite{nest_conv_comp}.\n\t\t\n\t\t\n\t\t \n\t}\n\\end{example}\n\n\\begin{example}[Nonlinear least squares]\\label{exa:nls}\n\t{\\rm\n\t\tThe composite problem class also captures nonlinear least squares problems with bound constraints:\n\t\t\\begin{align*}\n\t\t\\min_x~ \\|c(x)\\|_2\\quad \\textrm{subject to}\\quad l_i\\leq x_i\\leq u_i ~\\forall i.\n\t\t\\end{align*}\n\t\tSuch problems pervade engineering and scientific applications.\n\n\t}\n\\end{example}\n\n\n\n\\begin{example}[Exact penalty formulations]\\label{exa:ep}\n\t{\\rm\n\t\tConsider a nonlinear optimization problem:\n\t\t\\begin{align*}\n\t\t\\min_x~ \\{f(x): G(x)\\in \\mathcal{K}\\},\n\t\t\\end{align*}\n\t\twhere $f$ and $G$ are smooth maps and \n\t\t$\\mathcal{K}$ is a closed convex cone.\n\t\tAn accompanying {\\em penalty formulation} -- ubiquitous in nonlinear optimization\n\t\t-- takes the form \n\t\t$$\\min_x~ f(x)+\\lambda \\cdot {\\rm dist}_{\\mathcal{K}}(G(x)),$$\n\t\twhere ${\\rm dist}_{\\mathcal{K}}(\\cdot)$ is the  distance  to $\\mathcal{K}$ in some norm.\n\t\n\t\tHistorically, exact penalty formulations served as the  early motivation for the  class \\eqref{eqn:comp}.\t\n\t\t\n\t}\n\\end{example}\n\n\n\\begin{example}[Robust phase retrieval]\n\t{\\rm\n\tPhase retrieval is a common computational problem, with applications in diverse areas, such as imaging, X-ray crystallography, and speech processing. For simplicity, I will focus on the version of the problem over the reals.\nThe (real) phase retrieval problem seeks to determine a point $x$ satisfying the magnitude conditions, $$|\\langle a_i,x\\rangle|\\approx b_i\\quad \\textrm{for }i=1,\\ldots,m,$$ where $a_i\\in {\\mathbb R}^d$ and $b_i\\in{\\mathbb R}$ are given. Whenever there are gross outliers in the measurements $b_i$, the following robust formulation of the problem is appealing \\cite{eM,duchi_ruan_PR, proj_weak_dim}:\n$$\\min_x ~\\tfrac{1}{m}\\sum_{i=1}^m |\\langle a_i,x\\rangle^2-b_i^2|.$$\nClearly, this is an instance of \\eqref{eqn:comp}. For some recent perspectives on phase retrieval, see the survey \\cite{luke_news_views}. There are numerous recent  nonconvex approaches to phase retrieval, which rely on alternate problem formulations; e.g., \\cite{wirt_flow,rand_quad,phase_nonconv}.}\t\n\\end{example}\n\n\\begin{example}[Robust PCA]\n{\\rm\nIn robust principal component analysis, one seeks to identify sparse corruptions of a low-rank matrix \\cite{rob_cand,chand}. One typical example is image deconvolution,  where the low-rank structure  models the background of an image while the sparse corruption models the foreground. \n Formally, given a $m\\times n$ matrix $M$, the goal is to find a decomposition $M=L+S$, where $L$ is low-rank and $S$ is sparse. A common formulation of the problem reads:\n$$\\min_{U\\in {\\mathbb R}^{m\\times r},V\\in {\\mathbb R}^{n\\times r}}~ \\|UV^T-M\\|_1,$$ \nwhere $r$ is the target rank. }\n\\end{example}\n\n\n\n\\begin{example}[Censored $\\mathbb{Z}_2$ synchronization]\n\t{\\rm\nA synchronization problem over a graph is to estimate group elements $g_1,\\ldots, g_n$ from  pairwise products  $g_ig_j^{-1}$ over a set of edges $ij\\in E$. For a list of application of such problem see \\cite{ban_boum,ang_sing,abbe_band}, and references therein. A simple instance is $\\mathbb{Z}_2$ synchronization, corresponding to the group on two elements $\\{-1,+1\\}$. The popular problem of detecting communities in a network, within the Binary Stochastic Block Model (SBM), can be modeled using $\\mathbb{Z}_2$ synchronization. \n\nFormally, given a partially observed matrix $M$, the goal is to recover a vector $ \\theta\\in \\{\\pm 1\\}^d$, satisfying $M_{ij}\\approx \\theta_i \\theta_j$ for all $ij\\in E$. When the entries of $M$ are corrupted by adversarial sign flips, one can postulate the following formulation\n$$\\min_{\\theta\\in {\\mathbb R}^{d}}~ \\|P_{E}(\\theta\\theta^T-M)\\|_1,$$ \nwhere the operator $P_E$ records the entries indexed by the edge set $E$. Clearly, this is again an instance of \\eqref{eqn:comp}.\n}\n\\end{example}\n\n\t\n\n\n\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\\section{The proximally guided subgradient method}\nAs the first example of contemporary applications of the \nproximal point method,  consider the problem of minimizing the expectation:\\footnote{For simplicity of the exposition, the minimization problem is unconstrained. Simple constraints can be accommodated using a projection operation.}\n$$\\min_{x\\in {\\mathbb R}^d}~ F(x)=\\mathbb{E}_{\\zeta} f(x,\\zeta).$$\nHere, $\\zeta$ is a random variable, and the only access  to $F$ is by sampling $\\zeta$.\n It is difficult to overstate the importance of this problem class (often called {\\em stochastic approximation}) in large-scale optimization; see e.g. \\cite{BB,jordan}. \n\n\nWhen the problem is convex, the stochastic subgradient method \\cite{stochave,rob_mon,latest_subgrad} has strong theoretical guarantees and is often the method of choice. \nIn contrast, when applied to nonsmooth and nonconvex problems, the behavior of the method is poorly understood. The recent paper  \\cite{prox_guide_subgrad} shows how to use the proximal point method to guide the subgradient iterates in this broader setting, with rigorous guarantees.\n\n\nHenceforth, assume that the function $x\\mapsto f(x,\\zeta)$ is $\\rho$-weakly convex and $L$-Lipschitz for each $\\zeta$. Davis and Grimmer \\cite{prox_guide_subgrad} proposed the scheme outlined in Algorithm~\\ref{alg:proxguide}. \n\n\n\\begin{algorithm}\n\t\\KwData{$x_0\\in {\\mathbb R}^d$, $\\{j_t\\}\\subset\\mathbb{N}$, $\\{\\alpha_j\\}\\subset{\\mathbb R}_{++}$}\n\t\\For{t=0,\\ldots,T}{\n\tSet $y_0=x_t$\\;\n\t\\For{$j=0,\\ldots,j_t-2$}{\n\t\tSample $\\zeta$ and choose $v_j\\in\\partial (f(\\cdot,\\zeta)+\\rho\\|\\cdot-x_t\\|^2)(y_j)$\\;\n\t\t$y_{j+1}= y_j-\\alpha_jv_j$\n}\n$x_{t+1}= \\frac{1}{j_t}\\sum_{j=0}^{j_t-1}y_j$}\n\\caption{Proximally guided stochastic subgradient method}\\label{alg:proxguide}\n\\end{algorithm}\n\nThe method proceeds by applying a proximal point method with each subproblem approximately solved by a stochastic subgradient method. The intuition is that each proximal subproblem is $\\rho\/2$-strongly convex and therefore according to well-known results (e.g. \\cite{a_simp_app,Rakhlin_subgrad,hazan_subgrad,MR3353214}), the stochastic subgradient method should converge at the rate $O(\\frac{1}{T})$  on the subproblem, in expectation. This intuition is not quite correct because the objective function of the subproblem is not globally Lipschitz -- a key assumption for the $O(\\frac{1}{T})$ rate. Nonetheless, the authors show that warm-starting the subgradient method for each proximal subproblem with the current proximal iterate corrects this issue, yielding a favorable guarantees \\cite[Theorem 1]{prox_guide_subgrad}. \n\nTo describe the rate of convergence, set\n$j_t=t+\\lceil 648\\log(648)\\rceil$ and $\\alpha_j=\\tfrac{2}{\\rho(j+49)}$ in Algorithm~\\ref{alg:proxguide}. Then the scheme will generate an iterate $x$ satisfying \n$$\\mathbb{E}_{\\zeta}[\\|\\nabla F_{2\\rho}(x)\\|^2]\\leq \\varepsilon$$\nafter at most \n$$O\\left(\\frac{\\rho^2(F(x_0)-\\inf  F)^2}{\\varepsilon^2}+\\frac{L^4 \\log^{4}(\\varepsilon^{-1})}{\\varepsilon^2}\\right)$$\nsubgradient evaluations. This rate agrees with analogous guarantees for stochastic gradient methods for smooth nonconvex functions \\cite{gl_stoch}. \nIt is also worth noting that convex constraints on $x$ can be easily incorporated into Algorithm~\\ref{alg:proxguide} by introducing a nearest-point projection in the definition of $y_{j+1}$. \n\n\\section{The prox-linear algorithm}\t\nFor well-structured weakly convex problems, one can hope for faster numerical methods than the subgradient scheme. In this section, I will focus on the composite problem class \\eqref{eqn:comp}. To simplify the exposition, I will assume $L=1$, which can always be arranged by rescaling.\n\nSince composite functions are weakly convex, one could apply the proximal point method directly, while setting the parameter $\\nu\\leq\\beta^{-1}$. Even though the proximal subproblems are strongly convex, they are not in a form that is most amenable to convex optimization techniques. Indeed, most convex optimization algorithms are designed for minimizing a sum of a convex function and a composition of a convex function with a {\\em linear} map. This observation suggests introducing the following modification to the proximal-point algorithm. Given a current iterate $x_t$, the {\\em prox-linear method} sets\n\\begin{align*}\nx_{t+1}=\\operatornamewithlimits{argmin}_x \\{F(x;x_t)+\\tfrac{\\beta}{2}\\|x-x_t\\|^2\\},\n\\end{align*}\nwhere $F(x;y)$ is the local convex model \n$$F(x;y):=g(x)+h\\left(c(y)+\\nabla c(y)(x-y)\\right).$$\nIn other words, each proximal subproblem is approximated by linearizing the smooth map $c$ at the current iterate $x_t$.\n\nThe main advantage is that each subproblem is now a sum of a strongly convex function and a composition of a Lipschitz convex function with a linear map. A variety of methods utilizing this structure can be formally applied; e.g. smoothing \\cite{smooth_min_nonsmooth}, saddle-point \\cite{mprox,cp}, and interior point algorithms \\cite{nes_nem,wright_PD}. Which of these methods is practical depends on the specifics of the problem, such as the size and the cost of vector-matrix multiplications.\n\nIt is instructive to note that in the simplest setting of additive composite problems (Example~\\ref{exa:add_comp}), the prox-linear method reduces to the popular proximal-gradient algorithm or ISTA \\cite{smoothing_beckT}. For nonlinear least squares, the prox-linear method is a close variant of   Gauss-Newton. \n\nRecall that the step-size of the proximal point method provides a convenient stopping criteria, since it directly relates to the gradient of the Moreau envelope -- a smooth approximation of the objective function. Is there such an interpretation for the prox-linear method? This question is central, since termination criteria is not only used to stop the method but also to judge its efficiency and to compare against competing methods.\n\nThe answer is yes. Even though one can not evaluate the gradient $\\|\\nabla F_{\\frac{1}{2\\beta}}\\|$ directly,\nthe scaled step-size of the prox-linear method $$\\mathcal{G}(x):=\\beta(x_{t+1}-x_t)$$ is a good surrogate \\cite[Theorem 4.5]{prox_lin_paq}: \n$$\\tfrac{1}{4} \\|\\nabla F_{\\frac{1}{2\\beta}}(x)\\| \\leq \\|\\mathcal{G}(x)\\|\\leq 3\\|\\nabla F_{\\frac{1}{2\\beta}}(x)\\|.$$\nIn particular, the prox-linear method will find a point $x$ satisfying \n $\\|\\nabla F_{\\frac{1}{2\\beta}}(x)\\|^2\\leq\\varepsilon$ after at most $O\\left(\\frac{\\beta(F(x_0)-\\inf F)}{\\varepsilon}\\right)$ iterations. In the simplest setting when $g=0$ and $h(t)=t$, this rate reduces to the well-known convergence guarantee of gradient descent, which is black-box optimal for $C^1$-smooth nonconvex optimization \\cite{grad_desc_opt}. \n \nIt is worthwhile to note that a number of improvements to the basic prox-linear method were recently proposed. The authors of \\cite{composite_cart} discuss trust region variants and their complexity guarantees, while \\cite{duchi_ruan} propose stochastic extensions of the scheme and prove almost sure convergence. The paper \\cite{prox_lin_paq} discusses overall complexity guarantees when the convex subproblems can only be solved by first-order methods, and proposes an inertial variant of the scheme whose convergence guarantees automatically adapt to the near-convexity of the problem.\n\n\\subsection{Local rapid convergence}\n\nUnder typical regularity conditions, the prox-linear method  exhibits the same types of rapid convergence guarantees as the proximal point method. I will illustrate with two intuitive and widely used regularity conditions, yielding local linear and quadratic convergence, respectively.\n\n\\begin{defn}[\\cite{tilt}]{\\rm\n\tA local minimizer $\\bar x$ of $F$ is  {\\em $\\alpha$-tilt-stable}  if there exists $r>0$ such that the solution map\n$$M: v\\mapsto \\operatornamewithlimits{argmin}_{x\\in B_r(\\bar x)} \\left\\{ F(x)-\\langle v,x \\rangle\\right\\}$$\nis $1\/\\alpha$-Lipschitz around $0$ with $M(0)=\\bar x$. }\n\\end{defn}\n\nThis condition might seem unfamiliar to convex optimization specialist.\nThough not obvious, tilt-stability is equivalent to a uniform quadratic growth property and  a subtle localization of strong convexity of $F$. See \\cite{tilt_adrian} or \\cite{Dima_Ng} for more details on these equivalences. Under the tilt-stability assumption, the prox-linear method initialized sufficiently close to $\\bar x$ produces iterates that converge  at a linear rate $1-\\alpha\/\\beta$.\n\n\nThe second regularity condition models sharp growth of the function around the minimizer. Let $S$ be the set of all stationary points of $F$, meaning $x$ lies in $S$ if and only if the directional derivative $F'(x;v)$ is nonnegative in every direction $v\\in {\\mathbb R}^d$.\n\\begin{defn}[\\cite{weak_sharp}]{\\rm\n\tA local minimizer $\\bar x$ of $F$ is {\\em sharp}  if there exists $\\alpha>0$ and a neighborhood $\\mathcal{X}$ of $\\bar x$ such that \n$$F(x)\\geq  F({\\rm proj}_S(x))+c\\cdot {\\rm dist}(x,S)\\qquad\\forall x\\in \\mathcal{X}.$$}\n\\end{defn}\n\nUnder the sharpness condition,  the prox-linear method initialized sufficiently close to $\\bar x$ produces iterates that converge quadratically. \n\nFor well-structured problems, one can hope to justify the two regularity conditions above under statistical assumptions. The recent work of Duchi and Ruan on the phase retrieval problem \\cite{duchi_ruan_PR}  is an interesting recent example. Under mild statistical assumptions on the data generating mechanism,  sharpness  is assured with high probability. Therefore the  prox-linear method (and even subgradient methods \\cite{proj_weak_dim}) converge rapidly, when initialized within a constant relative distance of an optimal solution.\n\n\n\n\n\\section{Catalyst acceleration}\nThe final example concerns inertial acceleration in convex optimization.  Setting the groundwork, consider a $\\mu$-strongly convex function  $f$ with a  $\\beta$-Lipschitz gradient map $x\\mapsto \\nabla f(x)$. Classically,  gradient descent  will find a point $x$ satisfying $f(x)-\\min f<\\varepsilon$ after at most $$O\\left(\\frac{\\beta}{\\mu}\\ln(1\/\\varepsilon)\\right)$$ iterations. Accelerated gradient methods, beginning with Nesterov~\\cite{nest_orig}, equip the gradient descent method with an inertial correction. \nSuch methods have the much lower complexity guarantee $$O\\left(\\sqrt{\\frac{\\beta}{\\mu}}\\ln(1\/\\varepsilon)\\right),$$ which is optimal  within the first-order oracle model of computation \\cite{complexity}. \n\t\nIt is natural to ask which other methods, aside from gradient descent, can be ``accelerated''. For example, one may wish to accelerate coordinate descent or so-called variance reduced methods for finite sum problems; I will comment on the latter problem class shortly. \n\nOne appealing strategy relies on the proximal point method. G\\\"{u}ler in \\cite{gul_prox_acc} showed that the proximal point method itself can be equipped with inertial steps leading to improved convergence guarantees. Building on this work, Lin, Mairal, and Harchaoui \\cite{catalyst} explained how to derive the {\\em total} complexity guarantees for an inexact accelerated proximal point method that take into account the cost of applying\n an arbitrary linearly convergent algorithm $\\mathcal{M}$ to the subproblems. Their {\\em Catalyst acceleration} framework is summarized in Algorithm~\\ref{alg:catalyst}.\n\n\\begin{algorithm}\n\t\\KwData{$x_0\\in {\\mathbb R}^d$, $\\kappa>0$, algorithm $\\mathcal{M}$}\nSet $q= \\mu\/(\\mu+\\kappa)$, $\\alpha_0=\\sqrt{q}$, and $y_0=x_0$\\;\n\t\\For{t=0,\\ldots,T}{\n\tUse $\\mathcal{M}$ to approximately solve:\n\\begin{equation}\\label{eqn:prox_subprob}\n\tx_t\\approx\\operatornamewithlimits{argmin}_{x\\in {\\mathbb R}^d} \\left\\{F(x)+\\frac{\\kappa}{2}\\|x-y_{t-1}\\|^2\\right\\}.\\;\n\\end{equation}\n\nCompute $\\alpha_t\\in (0,1)$ from the equation $$\\alpha_t^2=(1-\\alpha_t)\\alpha_{t-1}^2+q\\alpha_t.\\;$$\n\nCompute:\n\\begin{align*}\n\\beta_t&=\\frac{\\alpha_{t-1}(1-\\alpha_{t-1})}{\\alpha_{t-1}^2+\\alpha_t},\\\\\ny_t&=x_t+\\beta_t(x_t-x_{t-1}). \n\\end{align*}\n}\n\\caption{Catalyst Acceleration}\\label{alg:catalyst}\n\\end{algorithm}\n\nTo state the guarantees of this method, suppose that $\\mathcal{M}$ converges on the proximal subproblem in function value at a linear rate $1-\\tau\\in (0,1)$. Then a simple termination policy on the subproblems \\eqref{eqn:prox_subprob} yields an algorithm with overall complexity \n\\begin{equation}\\label{eqn:compl}\n\\widetilde{O}\\left(\\frac{\\sqrt{\\mu+\\kappa}}{\\tau \\sqrt{\\mu}}\\ln(1\/\\varepsilon)\\right).\n\\end{equation}\nThat is, the expression \\eqref{eqn:compl} describes the maximal number of iterations of $\\mathcal{M}$ used by Algorithm~\\ref{alg:catalyst} until it finds a point $x$ satisfying $f(x)-\\inf f\\leq \\varepsilon$.\nTypically $\\tau$ depends on $\\kappa$; therefore the best choice of $\\kappa$ is the one that minimizes the ratio $\\frac{\\sqrt{\\mu+\\kappa}}{\\tau \\sqrt{\\mu}}$.\n\nThe main motivation for the Catalyst framework, and its most potent application, is the regularized Empirical Risk Minimization (ERM) problem:\n$$\\min_{x\\in {\\mathbb R}^d} f(x):=\\frac{1}{m}\\sum_{i=1}^m f_i(x)+g(x).$$\nSuch large-finite sum problems are ubiquitous in machine learning and high-dimensional statistics, where each function $f_i$ typically models a misfit between predicted and observed data while $g$ promotes some low dimensional structure on $x$, such as sparsity or low-rank.\n\nAssume that $f$ is $\\mu$-strongly convex and each individual $f_i$ is $ C^1$-smooth with $\\beta$-Lipschitz gradient. Since $m$ is assumed to be huge, the complexity of numerical methods is best measured in terms of the total number of individual gradient evaluations $\\nabla f_i$. In particular, fast gradient methods have the worst-case complexity $$O\\left(m\\sqrt{\\frac{\\beta}{\\mu}}\\ln(1\/\\varepsilon)\\right),$$ since each iteration requires evaluation of all the individual gradients $\\{\\nabla f_i(x)\\}_{i=1}^m$. Variance reduced algorithms, such as SAG \\cite{sag}, SAGA \\cite{SAGA2}, SDCA \\cite{sdca}, SMART \\cite{smart_davis}, SVRG \\cite{svrg,prox_SVRG}, FINITO \\cite{finito}, and MISO \\cite{miso,catalyst}, aim to improve the dependence on $m$. In their raw form, all of these methods exhibit a similar complexity $$O\\left(\\left(m+\\frac{\\beta}{\\mu}\\right)\\ln(1\/\\varepsilon)\\right),$$ in expectation, and differ only in storage requirements and in whether one needs to know explicitly the strong convexity constant. \n\nIt was a long standing open question to determine if the dependence on $\\beta\/\\mu$ can be improved. This is not quite possible in full generality, and instead one should expect a rate of the form\n $$O\\left(\\left(m+\\sqrt{m\\frac{\\beta}{\\mu}}\\right)\\ln(1\/\\varepsilon)\\right).$$ \nIndeed, such a rate would be optimal in an appropriate oracle model of complexity \\cite{NIPS2016_6058,yosi,AgB,conjugategradient}. Thus acceleration for ERM problems is only beneficial in the setting $m< \\beta\/\\mu$.\n  \n  \n  Early examples for specific algorithms are the accelerated SDCA \\cite{accsdca} and RPDG \\cite{conjugategradient}.\\footnote{Here, I am ignoring logarithmic terms in the convergence rate.} The accelerated SDCA, in particular, uses a specialized proximal-point construction and  was the motivation for the Catalyst framework. Catalyst generic acceleration allows to accelerate all of the variance reduced methods above in a single conceptually transparent framework.  It is worth noting that the first direct accelerated variance reduced methods for ERM problems were recently proposed in \\cite{accsvrg,NIPS2016_6154}. \n  \n  In contrast to the convex setting, the role of inertia for nonconvex problems is not nearly as well understood. In particular, gradient descent is black-box optimal for $C^1$-smooth nonconvex minimization \\cite{grad_desc_opt}, and therefore inertia can not help in the worst case. On the other hand, the recent paper \\cite{pmlr-v70-carmon17a} presents a first-order method for minimizing $C^2$ and $C^3$ smooth functions that is provably faster than gradient descent. At its core, their algorithm also combines inertia with the proximal point method.\n  For a partial extension of the Catalyst framework to weakly convex problems, see \\cite{catalyst_2}. \n \n  \n  \n\\section{Conclusion}\nThe proximal point method has long been ingrained in the foundations of optimization.  Recent  progress in large scale computing has shown that the proximal point method is not only conceptual, but can guide methodology. Though direct methods are usually preferable, proximally guided algorithms can be equally effective and often lead to more easily interpretable numerical methods.\nIn this article, I outlined three examples of this viewpoint, where the proximal-point method guides both the design and analysis of numerical methods.  \n\t\\bigskip\n\t\n\\noindent{\\bf Acknowledgments.}  The author thanks Damek Davis, John Duchi, and Zaid Harchaoui for their helpful comments on an early draft of the article. Research of Drusvyatskiy is supported by the AFOSR YIP award FA9550-15-1-0237 and by the NSF DMS   1651851 and CCF 1740551 awards.\t\n\t\n\t\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}